docs(mcp): document websearch provider configuration

src/mcp/AGENTS.md

@@ -25,7 +25,7 @@ mcp/
 
 | Name | URL | Purpose | Auth |
 |------|-----|---------|------|
-| websearch | mcp.exa.ai/mcp?tools=web_search_exa | Real-time web search | EXA_API_KEY |
+| websearch | mcp.exa.ai / mcp.tavily.com | Real-time web search | EXA_API_KEY / TAVILY_API_KEY |
 | context7 | mcp.context7.com/mcp | Library docs | CONTEXT7_API_KEY |
 | grep_app | mcp.grep.app | GitHub code search | None |
 
@@ -35,6 +35,36 @@ mcp/
 2. **Claude Code compat**: `.mcp.json` with `${VAR}` expansion
 3. **Skill-embedded**: YAML frontmatter in skills (handled by skill-mcp-manager)
 
+## Websearch Provider Configuration
+
+The `websearch` MCP supports multiple providers. Exa is the default for backward compatibility and works without an API key.
+
+| Provider | URL | Auth | API Key Required |
+|----------|-----|------|------------------|
+| exa (default) | mcp.exa.ai | x-api-key header | No (optional) |
+| tavily | mcp.tavily.com | Authorization Bearer | Yes |
+
+### Configuration Example
+
+```jsonc
+{
+  "websearch": {
+    "provider": "tavily"  // or "exa" (default)
+  }
+}
+```
+
+### Environment Variables
+
+- `EXA_API_KEY`: Optional. Used when provider is `exa`.
+- `TAVILY_API_KEY`: Required when provider is `tavily`.
+
+### Priority and Behavior
+
+- **Default**: Exa is used if no provider is specified.
+- **Backward Compatibility**: Existing setups using `EXA_API_KEY` continue to work without changes.
+- **Validation**: Selecting `tavily` without providing `TAVILY_API_KEY` will result in a configuration error.
+
 ## CONFIG PATTERN
 
 ```typescript
@@ -68,3 +98,4 @@ const mcps = createBuiltinMcps(["websearch"])  // Disable specific
 - **Disable**: User can set `disabled_mcps: ["name"]` in config
 - **Context7**: Optional auth using `CONTEXT7_API_KEY` env var
 - **Exa**: Optional auth using `EXA_API_KEY` env var
+- **Tavily**: Requires `TAVILY_API_KEY` env var