[GRO-1938] feat: Add Browserbase Eve extension

[Kylejeong2]

Summary

• add a publishable @browserbasehq/eve extension built from Eve primitives
• expose Stagehand navigation, observe, act, extract, and autonomous agent tools
• provide durable create/stop session tools that reconnect through Eve session state
• add Browserbase SDK-backed Search and Fetch tools
• add a runnable pnpm Eve example using Browserbase Model Gateway
• document Node 24 setup and the Search → Fetch → browser escalation path

Why

GRO-1938 calls for a Browserbase extension for Eve. This gives Eve agents one installable package for lightweight web discovery and retrieval plus persistent, interactive Browserbase sessions.

Stagehand is configured to use Browserbase Model Gateway, so consumers only need BROWSERBASE_API_KEY for browser sessions and Stagehand inference. The example uses Vercel AI Gateway for the outer Eve agent.

Validation

• pnpm check
• pnpm --filter browserbase-eve-example typecheck
• pnpm --filter browserbase-eve-example build
• live Stagehand observe smoke test through Browserbase Model Gateway
• live Browserbase SDK Search and Fetch smoke tests
• git diff --check

Linear: GRO-1938

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Note

Low Risk
New example-only packages under examples/integrations/vercel/ with no changes to production libraries or auth paths.

Overview
Adds a new Vercel integration example for Eve: a publishable @browserbasehq/eve package plus a sibling eve-example agent in a pnpm workspace.

The extension wires Browserbase Search/Fetch (no session) and Stagehand browser tools (navigate, observe, act, extract, autonomous agent) behind Eve tool namespaces. Keep-alive Browserbase sessions are stored in Eve durable state so tools reconnect across workflow steps; each call disconnects Stagehand without tearing down the remote browser until stop_session. Parallel tool calls in one step are serialized via a per-session lock, and session release is polled after REQUEST_RELEASE.

Stagehand uses Browserbase Model Gateway (only BROWSERBASE_API_KEY required for browser + inference). The example agent uses Vercel AI Gateway for the outer Eve model and documents Node 24 and a Search → Fetch → browser escalation path. The top-level Vercel examples README now lists this as the third example.

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