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/protocol/src/protocol.rs

@@ -437,6 +437,8 @@ pub enum EventMsg {
 
     McpToolCallEnd(McpToolCallEndEvent),
 
+    WebSearchBegin(WebSearchBeginEvent),
+
     /// Notification that the server is about to execute a command.
     ExecCommandBegin(ExecCommandBeginEvent),
 
@@ -658,6 +660,12 @@ impl McpToolCallEndEvent {
     }
 }
 
+#[derive(Debug, Clone, Deserialize, Serialize)]
+pub struct WebSearchBeginEvent {
+    pub call_id: String,
+    pub query: String,
+}
+
 /// Response payload for `Op::GetHistory` containing the current session's
 /// in-memory transcript.
 #[derive(Debug, Clone, Deserialize, Serialize)]