[Doc][ResponsesAPI] add documentation (#30840)

Signed-off-by: Andrew Xia <axia@fb.com>
Co-authored-by: Andrew Xia <axia@fb.com>

docs/serving/openai_compatible_server.md

@@ -47,6 +47,8 @@ We currently support the following OpenAI APIs:
 - [Completions API](#completions-api) (`/v1/completions`)
     - Only applicable to [text generation models](../models/generative_models.md).
     - *Note: `suffix` parameter is not supported.*
+- [Responses API](#responses-api) (`/v1/responses`)
+    - Only applicable to [text generation models](../models/generative_models.md).
 - [Chat Completions API](#chat-api) (`/v1/chat/completions`)
     - Only applicable to [text generation models](../models/generative_models.md) with a [chat template](../serving/openai_compatible_server.md#chat-template).
     - *Note: `user` parameter is ignored.*
@@ -229,6 +231,31 @@ The following extra parameters are supported:
     --8<-- "vllm/entrypoints/openai/protocol.py:chat-completion-extra-params"
     ```
 
+### Responses API
+
+Our Responses API is compatible with [OpenAI's Responses API](https://platform.openai.com/docs/api-reference/responses);
+you can use the [official OpenAI Python client](https://github.com/openai/openai-python) to interact with it.
+
+Code example: [examples/online_serving/openai_responses_client_with_tools.py](../../examples/online_serving/openai_responses_client_with_tools.py)
+
+#### Extra parameters
+
+The following extra parameters in the request object are supported:
+
+??? code
+
+    ```python
+    --8<-- "vllm/entrypoints/openai/protocol.py:responses-extra-params"
+    ```
+
+The following extra parameters in the response object are supported:
+
+??? code
+
+    ```python
+    --8<-- "vllm/entrypoints/openai/protocol.py:responses-response-extra-params"
+    ```
+
 ### Embeddings API
 
 Our Embeddings API is compatible with [OpenAI's Embeddings API](https://platform.openai.com/docs/api-reference/embeddings);