Add native Exa-backed web_search support (#29342)

Co-authored-by: Xinyuan Tong <115166877+JustinTong0323@users.noreply.github.com>

python/sglang/srt/entrypoints/http_server.py

@@ -337,6 +337,10 @@ async def lifespan(fast_api_app: FastAPI):
 
         tool_server = MCPToolServer()
         await tool_server.add_tool_server(server_args.tool_server)
+    elif envs.EXA_API_KEY.get():
+        from sglang.srt.entrypoints.openai.tool_server import NativeToolServer
+
+        tool_server = NativeToolServer()
 
     try:
         from sglang.srt.entrypoints.openai.serving_responses import (
@@ -381,6 +385,8 @@ async def lifespan(fast_api_app: FastAPI):
     try:
         yield
     finally:
+        if tool_server is not None and hasattr(tool_server, "aclose"):
+            await tool_server.aclose()
         warmup_thread.join()