Allow custom providers to opt into standalone web search (#34846)

## What changed

- Add the `supports_standalone_web_search` model-provider setting, defaulting to `false`, and preserve it in remote thread configuration.
- Enable the standalone `web.run` tool for opted-in custom Responses providers when web search is enabled and the runtime provider supports it.
- Send standalone search requests through the custom provider's endpoint and authentication.

## Testing

- Cover opt-in, default-off, disabled-search, and unsupported-provider behavior.
- Verify custom-provider search request routing and authorization through the app server.

GitOrigin-RevId: 7c5f96b0ce924ad2b9715c45bfc635e89fc39cff

codex-rs/core/src/compact_tests.rs

@@ -284,6 +284,7 @@ fn should_use_remote_compact_task_for_azure_provider() {
         websocket_connect_timeout_ms: None,
         requires_openai_auth: false,
         supports_websockets: false,
+        supports_standalone_web_search: false,
     };
 
     assert!(should_use_remote_compact_task(&provider));