Codex: add native web search for embedded Pi runs

docs/tools/web.md

@@ -102,6 +102,43 @@ local while `web_search` and `x_search` can use xAI Responses under the hood.
 
 ## Auto-detection
 
+## Native Codex web search
+
+Codex-capable models can optionally use the provider-native Responses `web_search` tool instead of OpenClaw's managed `web_search` function.
+
+- Configure it under `tools.web.search.openaiCodex`
+- It only activates for Codex-capable models (`openai-codex/*` or providers using `api: "openai-codex-responses"`)
+- Managed `web_search` still applies to non-Codex models
+- `mode: "cached"` is the default and recommended setting
+- `tools.web.search.enabled: false` disables both managed and native search
+
+```json5
+{
+  tools: {
+    web: {
+      search: {
+        enabled: true,
+        openaiCodex: {
+          enabled: true,
+          mode: "cached",
+          allowedDomains: ["example.com"],
+          contextSize: "high",
+          userLocation: {
+            country: "US",
+            city: "New York",
+            timezone: "America/New_York",
+          },
+        },
+      },
+    },
+  },
+}
+```
+
+If native Codex search is enabled but the current model is not Codex-capable, OpenClaw keeps the normal managed `web_search` behavior.
+
+## Setting up web search
+
 Provider lists in docs and setup flows are alphabetical. Auto-detection keeps a
 separate precedence order: