fix: dont auto-enable web_search for azure (#10266)

seeing issues with azure after default-enabling web search: #10071,
#10257.

need to work with azure to fix api-side, for now turning off
default-enable of web_search for azure.

diff is big because i moved logic to reuse

codex-rs/core/tests/suite/mod.rs

@@ -80,5 +80,5 @@ mod unstable_features_warning;
 mod user_notification;
 mod user_shell_cmd;
 mod view_image;
-mod web_search_cached;
+mod web_search;
 mod websocket_fallback;

codex-rs/core/tests/suite/web_search.rs

@@ -1,5 +1,7 @@
 #![allow(clippy::unwrap_used)]
 
+use codex_core::WireApi;
+use codex_core::built_in_model_providers;
 use codex_core::features::Feature;
 use codex_core::protocol::SandboxPolicy;
 use codex_protocol::config_types::WebSearchMode;
@@ -25,6 +27,15 @@ fn find_web_search_tool(body: &Value) -> &Value {
         .expect("tools should include a web_search tool")
 }
 
+#[allow(clippy::expect_used)]
+fn has_web_search_tool(body: &Value) -> bool {
+    body["tools"]
+        .as_array()
+        .expect("request body should include tools array")
+        .iter()
+        .any(|tool| tool.get("type").and_then(Value::as_str) == Some("web_search"))
+}
+
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
 async fn web_search_mode_cached_sets_external_web_access_false() {
     skip_if_no_network!();
@@ -174,3 +185,45 @@ async fn web_search_mode_updates_between_turns_with_sandbox_policy() {
         "danger-full-access policy should default web_search to live"
     );
 }
+
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn web_search_mode_defaults_to_disabled_for_azure_responses() {
+    skip_if_no_network!();
+
+    let server = start_mock_server().await;
+    let sse = sse_completed("resp-1");
+    let resp_mock = responses::mount_sse_once(&server, sse).await;
+
+    let mut builder = test_codex()
+        .with_model("gpt-5-codex")
+        .with_config(|config| {
+            let base_url = config.model_provider.base_url.clone();
+            let mut provider = built_in_model_providers()["openai"].clone();
+            provider.name = "Azure".to_string();
+            provider.base_url = base_url;
+            provider.wire_api = WireApi::Responses;
+            config.model_provider_id = provider.name.clone();
+            config.model_provider = provider;
+            config.web_search_mode = None;
+            config.features.disable(Feature::WebSearchCached);
+            config.features.disable(Feature::WebSearchRequest);
+        });
+    let test = builder
+        .build(&server)
+        .await
+        .expect("create test Codex conversation");
+
+    test.submit_turn_with_policy(
+        "hello azure default web search",
+        SandboxPolicy::DangerFullAccess,
+    )
+    .await
+    .expect("submit turn");
+
+    let body = resp_mock.single_request().body_json();
+    assert_eq!(
+        has_web_search_tool(&body),
+        false,
+        "azure responses requests should disable web_search by default"
+    );
+}