[Feature][Response API] Add streaming support for non-harmony (#23741)

Signed-off-by: Kebe <mail@kebe7jun.com>

tests/v1/entrypoints/openai/responses/test_basic.py

@@ -2,6 +2,7 @@
 # SPDX-FileCopyrightText: Copyright contributors to the vLLM project
 
 import openai  # use the official client for correctness check
+import openai.types.responses as openai_responses_types
 import pytest
 
 
@@ -86,3 +87,18 @@ async def test_logprobs(client: openai.AsyncOpenAI):
     outputs = response.output
     assert outputs[-1].content[-1].logprobs
     assert len(outputs[-1].content[-1].logprobs[0].top_logprobs) == 5
+
+
+@pytest.mark.asyncio
+async def test_streaming(client: openai.AsyncOpenAI):
+    stream = await client.responses.create(
+        input="What is 13 * 24?",
+        stream=True,
+    )
+    events = [event async for event in stream]
+    assert isinstance(events[0], openai_responses_types.ResponseCreatedEvent)
+    assert any(
+        isinstance(event, openai_responses_types.ResponseTextDeltaEvent)
+        for event in events)
+    assert isinstance(events[-1],
+                      openai_responses_types.ResponseCompletedEvent)