docs(cli): add Vercel provider to setup and doctor

src/cli/doctor.test.ts

@@ -331,6 +331,37 @@ models:
     }
   });
 
+  it('reports WARN when Vercel AI Gateway has no available API key sources', async () => {
+    const originalKey = process.env.AI_GATEWAY_API_KEY;
+    delete process.env.AI_GATEWAY_API_KEY;
+
+    try {
+      mkdirSync(testDir, { recursive: true });
+      const configPath = join(testDir, 'vercel-missing.yaml');
+      writeFileSync(configPath, `
+telegram:
+  bot_token: "test-token"
+  allowed_chat_ids: [123]
+models:
+  default:
+    provider: vercel
+    model: openai/gpt-4.1
+`);
+
+      const ctx: DoctorContext = { configPath, dataDir: testDir };
+      const results = await runChecks(ctx);
+      const modelCheck = results.find(r => r.label.includes('Model connectivity')) as CheckResult | undefined;
+      expect(modelCheck?.status).toBe('warn');
+      expect(modelCheck?.detail).toContain('AI_GATEWAY_API_KEY');
+    } finally {
+      if (originalKey) {
+        process.env.AI_GATEWAY_API_KEY = originalKey;
+      } else {
+        delete process.env.AI_GATEWAY_API_KEY;
+      }
+    }
+  });
+
   it('reports FAIL when Anthropic auth_mode=oauth has no available auth token sources', async () => {
     const originalHome = process.env.HOME;
     const originalToken = process.env.ANTHROPIC_AUTH_TOKEN;