[chatbot-demo]: Add Vercel serverless deployment config

[jonah-berman]

Summary

Converts the Express.js backend to Vercel serverless functions to enable deployment on Vercel. Creates two API endpoints (/api/chat and /api/chat/stream) that replicate the existing server.js functionality, plus a vercel.json configuration file.

The serverless functions include all the Exa search logic, OpenRouter LLM integration, and system prompts inline (duplicated from the original server.js/exa.js/prompt.js files).

Review & Testing Checklist for Human

• Configure environment variables in Vercel: OPEN_ROUTER_KEY and EXA_API_KEY must be set in the Vercel project settings before the deployment will work
• Test SSE streaming: Verify that /api/chat/stream correctly streams responses - SSE in Vercel serverless can behave differently than Express
• Verify maxDuration: The 60s function timeout requires Vercel Pro plan; if on Hobby plan, functions timeout at 10s which may not be enough for LLM + search operations
• Test the deployed app end-to-end: Send a chat message, verify Exa search triggers, and confirm the response streams back correctly

Notes

The code in api/chat.js and api/chat/stream.js is largely duplicated - this was done intentionally to keep the serverless functions self-contained, but could be refactored to share common modules if this becomes a maintenance burden.

Requested by: @jonah-berman
Link to Devin run: https://app.devin.ai/sessions/e718255bc0da453a9e25487a80118e85

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