feat(vercel): add Stagehand code-mode binding

[shrey150]

Summary

• add a native Vercel AI SDK code_execute tool backed by the shared Stagehand executor
• add an optional ai peer and a dedicated ./ai-sdk package export
• load the shared Stagehand V4 syntax guide into the example agent's system context
• add a runnable Vercel AI SDK example, discovery smoke, and isolated CI job

Stack

This PR is stacked on the framework-neutral code-mode tool. Its diff contains only the Vercel AI SDK binding and example.

Lifecycle

One native tool instance owns one executor for the complete generateText call. The example closes that executor in finally, ensuring its Browserbase browser is released whether generation succeeds or fails. The native binding hosts execution directly instead of launching a local MCP subprocess.

E2E Test Matrix

Command / flow	Observed output	Confidence / sufficiency
pnpm --filter @browserbasehq/stagehand-codemode run build followed by pnpm --filter stagehand-codemode-ai-sdk-example run smoke	The local package built its ./ai-sdk export and the example reported Vercel AI SDK native tool binding PASS: code_execute.	Proves the example imports the exact local build and receives the expected native tool name, schema, executor callback, and shared syntax guide. It does not make a provider-dependent model request.
pnpm --filter stagehand-codemode-ai-sdk-example run check	TypeScript exited successfully.	Proves the example compiles against the reviewed local binding and current AI SDK types.
pnpm --filter @browserbasehq/stagehand-codemode run typecheck	TypeScript exited successfully.	Confirms adding the optional AI SDK export remains compatible with the shared executor and MCP types.
Targeted ESLint and Prettier checks over the package, example, workspace, and workflow	Both completed without warnings or formatting differences.	Covers the complete changed TypeScript, JavaScript, JSON, YAML, and Markdown surface.

[shrey150]

Superseded by browserbase/stagehand#2626, which keeps the Vercel AI SDK example beside the canonical MCP build and tests both local and Browserbase startup.