[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

Medium Risk
New external integration surface (API keys, billable browser sessions, session release correctness) and session concurrency logic; scope is additive (new package + example) with no changes to existing production paths.

Overview
Introduces @browserbasehq/eve, a publishable Eve extension that wires Browserbase Search/Fetch and Stagehand browser tools into agent workflows. Agents get namespaced tools for search, fetch, create/stop session, navigate, observe, act, extract, and multi-step agent runs.

Persistent sessions are tracked in Eve durable state (browser-session); each tool reconnects via keep-alive Browserbase sessions, disconnects Stagehand after one operation, and stop_session requests release with polling. Parallel tool calls in a single step are serialized with an in-memory session lock; stale sessions are cleared when the remote session is terminal.

Stagehand uses Browserbase Model Gateway (only BROWSERBASE_API_KEY required for browser + inference). Config covers model, timeout, and proxies.

Adds a pnpm workspace linking packages/eve-browserbase to examples/integrations/vercel/eve-example (runnable Eve agent, env template, Node 24). The Vercel integrations README now documents this as a third example.

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