chore: update the vLLM inference impl to use OpenAIMixin for openai-compat functions (#3404)

# What does this PR do?

update vLLM inference provider to use OpenAIMixin for openai-compat
functions

inference recordings from Qwen3-0.6B and vLLM 0.8.3 -
```
docker run --gpus all -v ~/.cache/huggingface:/root/.cache/huggingface -p 8000:8000 --ipc=host \
    vllm/vllm-openai:latest \
    --model Qwen/Qwen3-0.6B --enable-auto-tool-choice --tool-call-parser hermes
```

## Test Plan

```
./scripts/integration-tests.sh --stack-config server:ci-tests --setup vllm --subdirs inference
```

llama_stack/providers/remote/inference/vllm/vllm.py

@@ -4,7 +4,7 @@
 # This source code is licensed under the terms described in the LICENSE file in
 # the root directory of this source tree.
 import json
-from collections.abc import AsyncGenerator, AsyncIterator
+from collections.abc import AsyncGenerator
 from typing import Any
 
 import httpx
@@ -38,13 +38,6 @@ from llama_stack.apis.inference import (
     LogProbConfig,
     Message,
     ModelStore,
-    OpenAIChatCompletion,
-    OpenAICompletion,
-    OpenAIEmbeddingData,
-    OpenAIEmbeddingsResponse,
-    OpenAIEmbeddingUsage,
-    OpenAIMessageParam,
-    OpenAIResponseFormatParam,
     ResponseFormat,
     SamplingParams,
     TextTruncation,
@@ -71,11 +64,11 @@ from llama_stack.providers.utils.inference.openai_compat import (
     convert_message_to_openai_dict,
     convert_tool_call,
     get_sampling_options,
-    prepare_openai_completion_params,
     process_chat_completion_stream_response,
     process_completion_response,
     process_completion_stream_response,
 )
+from llama_stack.providers.utils.inference.openai_mixin import OpenAIMixin
 from llama_stack.providers.utils.inference.prompt_adapter import (
     completion_request_to_prompt,
     content_has_media,
@@ -288,7 +281,7 @@ async def _process_vllm_chat_completion_stream_response(
         yield c
 
 
-class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
+class VLLMInferenceAdapter(OpenAIMixin, Inference, ModelsProtocolPrivate):
     # automatically set by the resolver when instantiating the provider
     __provider_id__: str
     model_store: ModelStore | None = None
@@ -296,7 +289,6 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
     def __init__(self, config: VLLMInferenceAdapterConfig) -> None:
         self.register_helper = ModelRegistryHelper(build_hf_repo_model_entries())
         self.config = config
-        self.client = None
 
     async def initialize(self) -> None:
         if not self.config.url:
@@ -308,8 +300,6 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
         return self.config.refresh_models
 
     async def list_models(self) -> list[Model] | None:
-        self._lazy_initialize_client()
-        assert self.client is not None  # mypy
         models = []
         async for m in self.client.models.list():
             model_type = ModelType.llm  # unclear how to determine embedding vs. llm models
@@ -340,8 +330,7 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
             HealthResponse: A dictionary containing the health status.
         """
         try:
-            client = self._create_client() if self.client is None else self.client
-            _ = [m async for m in client.models.list()]  # Ensure the client is initialized
+            _ = [m async for m in self.client.models.list()]  # Ensure the client is initialized
             return HealthResponse(status=HealthStatus.OK)
         except Exception as e:
             return HealthResponse(status=HealthStatus.ERROR, message=f"Health check failed: {str(e)}")
@@ -351,19 +340,14 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
             raise ValueError("Model store not set")
         return await self.model_store.get_model(model_id)
 
-    def _lazy_initialize_client(self):
-        if self.client is not None:
-            return
+    def get_api_key(self):
+        return self.config.api_token
 
-        log.info(f"Initializing vLLM client with base_url={self.config.url}")
-        self.client = self._create_client()
+    def get_base_url(self):
+        return self.config.url
 
-    def _create_client(self):
-        return AsyncOpenAI(
-            base_url=self.config.url,
-            api_key=self.config.api_token,
-            http_client=httpx.AsyncClient(verify=self.config.tls_verify),
-        )
+    def get_extra_client_params(self):
+        return {"http_client": httpx.AsyncClient(verify=self.config.tls_verify)}
 
     async def completion(
         self,
@@ -374,7 +358,6 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
         stream: bool | None = False,
         logprobs: LogProbConfig | None = None,
     ) -> CompletionResponse | AsyncGenerator[CompletionResponseStreamChunk, None]:
-        self._lazy_initialize_client()
         if sampling_params is None:
             sampling_params = SamplingParams()
         model = await self._get_model(model_id)
@@ -406,7 +389,6 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
         logprobs: LogProbConfig | None = None,
         tool_config: ToolConfig | None = None,
     ) -> ChatCompletionResponse | AsyncGenerator[ChatCompletionResponseStreamChunk, None]:
-        self._lazy_initialize_client()
         if sampling_params is None:
             sampling_params = SamplingParams()
         model = await self._get_model(model_id)
@@ -479,16 +461,12 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
             yield chunk
 
     async def register_model(self, model: Model) -> Model:
-        # register_model is called during Llama Stack initialization, hence we cannot init self.client if not initialized yet.
-        # self.client should only be created after the initialization is complete to avoid asyncio cross-context errors.
-        # Changing this may lead to unpredictable behavior.
-        client = self._create_client() if self.client is None else self.client
         try:
             model = await self.register_helper.register_model(model)
         except ValueError:
             pass  # Ignore statically unknown model, will check live listing
         try:
-            res = await client.models.list()
+            res = await self.client.models.list()
         except APIConnectionError as e:
             raise ValueError(
                 f"Failed to connect to vLLM at {self.config.url}. Please check if vLLM is running and accessible at that URL."
@@ -543,8 +521,6 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
         output_dimension: int | None = None,
         task_type: EmbeddingTaskType | None = None,
     ) -> EmbeddingsResponse:
-        self._lazy_initialize_client()
-        assert self.client is not None
         model = await self._get_model(model_id)
 
         kwargs = {}
@@ -560,154 +536,3 @@ class VLLMInferenceAdapter(Inference, ModelsProtocolPrivate):
 
         embeddings = [data.embedding for data in response.data]
         return EmbeddingsResponse(embeddings=embeddings)
-
-    async def openai_embeddings(
-        self,
-        model: str,
-        input: str | list[str],
-        encoding_format: str | None = "float",
-        dimensions: int | None = None,
-        user: str | None = None,
-    ) -> OpenAIEmbeddingsResponse:
-        self._lazy_initialize_client()
-        assert self.client is not None
-        model_obj = await self._get_model(model)
-        assert model_obj.model_type == ModelType.embedding
-
-        # Convert input to list if it's a string
-        input_list = [input] if isinstance(input, str) else input
-
-        # Call vLLM embeddings endpoint with encoding_format
-        response = await self.client.embeddings.create(
-            model=model_obj.provider_resource_id,
-            input=input_list,
-            dimensions=dimensions,
-            encoding_format=encoding_format,
-        )
-
-        # Convert response to OpenAI format
-        data = [
-            OpenAIEmbeddingData(
-                embedding=embedding_data.embedding,
-                index=i,
-            )
-            for i, embedding_data in enumerate(response.data)
-        ]
-
-        # Not returning actual token usage since vLLM doesn't provide it
-        usage = OpenAIEmbeddingUsage(prompt_tokens=-1, total_tokens=-1)
-
-        return OpenAIEmbeddingsResponse(
-            data=data,
-            model=model_obj.provider_resource_id,
-            usage=usage,
-        )
-
-    async def openai_completion(
-        self,
-        model: str,
-        prompt: str | list[str] | list[int] | list[list[int]],
-        best_of: int | None = None,
-        echo: bool | None = None,
-        frequency_penalty: float | None = None,
-        logit_bias: dict[str, float] | None = None,
-        logprobs: bool | None = None,
-        max_tokens: int | None = None,
-        n: int | None = None,
-        presence_penalty: float | None = None,
-        seed: int | None = None,
-        stop: str | list[str] | None = None,
-        stream: bool | None = None,
-        stream_options: dict[str, Any] | None = None,
-        temperature: float | None = None,
-        top_p: float | None = None,
-        user: str | None = None,
-        guided_choice: list[str] | None = None,
-        prompt_logprobs: int | None = None,
-        suffix: str | None = None,
-    ) -> OpenAICompletion:
-        self._lazy_initialize_client()
-        model_obj = await self._get_model(model)
-
-        extra_body: dict[str, Any] = {}
-        if prompt_logprobs is not None and prompt_logprobs >= 0:
-            extra_body["prompt_logprobs"] = prompt_logprobs
-        if guided_choice:
-            extra_body["guided_choice"] = guided_choice
-
-        params = await prepare_openai_completion_params(
-            model=model_obj.provider_resource_id,
-            prompt=prompt,
-            best_of=best_of,
-            echo=echo,
-            frequency_penalty=frequency_penalty,
-            logit_bias=logit_bias,
-            logprobs=logprobs,
-            max_tokens=max_tokens,
-            n=n,
-            presence_penalty=presence_penalty,
-            seed=seed,
-            stop=stop,
-            stream=stream,
-            stream_options=stream_options,
-            temperature=temperature,
-            top_p=top_p,
-            user=user,
-            extra_body=extra_body,
-        )
-        return await self.client.completions.create(**params)  # type: ignore
-
-    async def openai_chat_completion(
-        self,
-        model: str,
-        messages: list[OpenAIMessageParam],
-        frequency_penalty: float | None = None,
-        function_call: str | dict[str, Any] | None = None,
-        functions: list[dict[str, Any]] | None = None,
-        logit_bias: dict[str, float] | None = None,
-        logprobs: bool | None = None,
-        max_completion_tokens: int | None = None,
-        max_tokens: int | None = None,
-        n: int | None = None,
-        parallel_tool_calls: bool | None = None,
-        presence_penalty: float | None = None,
-        response_format: OpenAIResponseFormatParam | None = None,
-        seed: int | None = None,
-        stop: str | list[str] | None = None,
-        stream: bool | None = None,
-        stream_options: dict[str, Any] | None = None,
-        temperature: float | None = None,
-        tool_choice: str | dict[str, Any] | None = None,
-        tools: list[dict[str, Any]] | None = None,
-        top_logprobs: int | None = None,
-        top_p: float | None = None,
-        user: str | None = None,
-    ) -> OpenAIChatCompletion | AsyncIterator[OpenAIChatCompletionChunk]:
-        self._lazy_initialize_client()
-        model_obj = await self._get_model(model)
-        params = await prepare_openai_completion_params(
-            model=model_obj.provider_resource_id,
-            messages=messages,
-            frequency_penalty=frequency_penalty,
-            function_call=function_call,
-            functions=functions,
-            logit_bias=logit_bias,
-            logprobs=logprobs,
-            max_completion_tokens=max_completion_tokens,
-            max_tokens=max_tokens,
-            n=n,
-            parallel_tool_calls=parallel_tool_calls,
-            presence_penalty=presence_penalty,
-            response_format=response_format,
-            seed=seed,
-            stop=stop,
-            stream=stream,
-            stream_options=stream_options,
-            temperature=temperature,
-            tool_choice=tool_choice,
-            tools=tools,
-            top_logprobs=top_logprobs,
-            top_p=top_p,
-            user=user,
-        )
-        return await self.client.chat.completions.create(**params)  # type: ignore