chore: update Vercel AI SDK implementation to support latest version

[nicoalbanese]

Problem

This project was using an old version of the Vercel AI SDK and did not take advantage of AI SDK Core's unified API.

Solution

This PR updates the AI implementation to use the latest version of the Vercel AI SDK, namely using the streamText function in the api/chat/route.ts rather than the OpenAI client. I also updated the README to reflect the updated implementation.

Type of Change

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• This change requires a documentation update
• Infrastructure change (CI configs, etc)
• Non-code change (docs, etc)
• None of the above: (explain here)

Test Plan

Tests pass.