fix(mypy): resolve OpenAI SDK and provider type issues (#3936)

## Summary
- Fix OpenAI SDK NotGiven/Omit type mismatches in embeddings calls
- Fix incorrect OpenAIChatCompletionChunk import in vllm provider
- Refactor to avoid type:ignore comments by using conditional kwargs

## Changes
**openai_mixin.py (9 errors fixed):**
- Build kwargs conditionally for embeddings.create() to avoid
NotGiven/Omit mismatch
- Only include parameters when they have actual values (not None)

**gemini.py (9 errors fixed):**
- Apply same conditional kwargs pattern
- Add missing Any import

**vllm.py (2 errors fixed):**
- Use correct OpenAIChatCompletionChunk from llama_stack.apis.inference
- Remove incorrect alias from openai package

## Technical Notes
The OpenAI SDK has a type system quirk where `NOT_GIVEN` has type
`NotGiven` but parameter signatures expect `Omit`. By only passing
parameters with actual values, we avoid this mismatch entirely without
needing `# type: ignore` comments.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

---------

Co-authored-by: Claude <noreply@anthropic.com>

src/llama_stack/providers/remote/inference/gemini/gemini.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.
 
-from openai import NOT_GIVEN
+from typing import Any
 
 from llama_stack.apis.inference import (
     OpenAIEmbeddingData,
@@ -37,21 +37,20 @@ class GeminiInferenceAdapter(OpenAIMixin):
         Override embeddings method to handle Gemini's missing usage statistics.
         Gemini's embedding API doesn't return usage information, so we provide default values.
         """
-        # Prepare request parameters
-        request_params = {
+        # Build request params conditionally to avoid NotGiven/Omit type mismatch
+        request_params: dict[str, Any] = {
             "model": await self._get_provider_model_id(params.model),
             "input": params.input,
-            "encoding_format": params.encoding_format if params.encoding_format is not None else NOT_GIVEN,
-            "dimensions": params.dimensions if params.dimensions is not None else NOT_GIVEN,
-            "user": params.user if params.user is not None else NOT_GIVEN,
         }
+        if params.encoding_format is not None:
+            request_params["encoding_format"] = params.encoding_format
+        if params.dimensions is not None:
+            request_params["dimensions"] = params.dimensions
+        if params.user is not None:
+            request_params["user"] = params.user
+        if params.model_extra:
+            request_params["extra_body"] = params.model_extra
 
-        # Add extra_body if present
-        extra_body = params.model_extra
-        if extra_body:
-            request_params["extra_body"] = extra_body
-
-        # Call OpenAI embeddings API with properly typed parameters
         response = await self.client.embeddings.create(**request_params)
 
         data = []

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

@@ -7,13 +7,11 @@ from collections.abc import AsyncIterator
 from urllib.parse import urljoin
 
 import httpx
-from openai.types.chat.chat_completion_chunk import (
-    ChatCompletionChunk as OpenAIChatCompletionChunk,
-)
 from pydantic import ConfigDict
 
 from llama_stack.apis.inference import (
     OpenAIChatCompletion,
+    OpenAIChatCompletionChunk,
     OpenAIChatCompletionRequestWithExtraBody,
     ToolChoice,
 )