Add web search tool (#2371)

Adds web_search tool, enabling the model to use Responses API web_search
tool.
- Disabled by default, enabled by --search flag
- When --search is passed, exposes web_search_request function tool to
the model, which triggers user approval. When approved, the model can
use the web_search tool for the remainder of the turn
<img width="1033" height="294" alt="image"
src="https://github.com/user-attachments/assets/62ac6563-b946-465c-ba5d-9325af28b28f"
/>

---------

Co-authored-by: easong-openai <easong@openai.com>

codex-rs/core/src/codex.rs

@@ -96,6 +96,7 @@ use crate::protocol::StreamErrorEvent;
 use crate::protocol::Submission;
 use crate::protocol::TaskCompleteEvent;
 use crate::protocol::TurnDiffEvent;
+use crate::protocol::WebSearchBeginEvent;
 use crate::rollout::RolloutRecorder;
 use crate::safety::SafetyCheck;
 use crate::safety::assess_command_safety;
@@ -511,6 +512,7 @@ impl Session {
                 sandbox_policy.clone(),
                 config.include_plan_tool,
                 config.include_apply_patch_tool,
+                config.tools_web_search_request,
                 config.use_experimental_streamable_shell_tool,
             ),
             user_instructions,
@@ -1096,6 +1098,7 @@ async fn submission_loop(
                     new_sandbox_policy.clone(),
                     config.include_plan_tool,
                     config.include_apply_patch_tool,
+                    config.tools_web_search_request,
                     config.use_experimental_streamable_shell_tool,
                 );
 
@@ -1175,6 +1178,7 @@ async fn submission_loop(
                             sandbox_policy.clone(),
                             config.include_plan_tool,
                             config.include_apply_patch_tool,
+                            config.tools_web_search_request,
                             config.use_experimental_streamable_shell_tool,
                         ),
                         user_instructions: turn_context.user_instructions.clone(),
@@ -1687,6 +1691,7 @@ async fn try_run_turn(
     let mut stream = turn_context.client.clone().stream(&prompt).await?;
 
     let mut output = Vec::new();
+
     loop {
         // Poll the next item from the model stream. We must inspect *both* Ok and Err
         // cases so that transient stream failures (e.g., dropped SSE connection before
@@ -1723,6 +1728,16 @@ async fn try_run_turn(
                 .await?;
                 output.push(ProcessedResponseItem { item, response });
             }
+            ResponseEvent::WebSearchCallBegin { call_id, query } => {
+                let q = query.unwrap_or_else(|| "Searching Web...".to_string());
+                let _ = sess
+                    .tx_event
+                    .send(Event {
+                        id: sub_id.to_string(),
+                        msg: EventMsg::WebSearchBegin(WebSearchBeginEvent { call_id, query: q }),
+                    })
+                    .await;
+            }
             ResponseEvent::Completed {
                 response_id: _,
                 token_usage,