docs: Add AI_GATEWAY_API_KEY to Vercel deployment guide

[inkeep]

Summary

Updates the Vercel deployment documentation to include the new AI_GATEWAY_API_KEY environment variable introduced in #2803.

Changes

• Added AI_GATEWAY_API_KEY to the Run API environment variables code block
• Added documentation in the environment variables table explaining that this optional variable enables automatic routing of Anthropic, OpenAI, and Google models through the Vercel AI Gateway for per-request cost tracking

Related

• Follows Route LLM calls through AI Gateway for per-request cost tracking #2803 "Route LLM calls through AI Gateway for per-request cost tracking"

---

This PR was automatically generated to document the AI Gateway cost tracking feature.

[vercel]

The latest updates on your projects. Learn more about Vercel for GitHub.

Project	Deployment	Actions	Updated (UTC)
agents-api	Ready	Preview, Comment	Mar 25, 2026 8:41pm
agents-docs	Ready	Preview, Comment	Mar 25, 2026 8:41pm
agents-manage-ui	Ready	Preview, Comment	Mar 25, 2026 8:41pm

[changeset-bot]

⚠️ No Changeset found

Latest commit: 2cc51c1

Merging this PR will not cause a version bump for any packages. If these changes should not result in a new version, you're good to go. If these changes should result in a version bump, you need to add a changeset.

This PR includes no changesets

When changesets are added to this PR, you'll see the packages that this PR includes changesets for and the associated semver types

Click here to learn what changesets are, and how to add one.

Click here if you're a maintainer who wants to add a changeset to this PR

[github-actions]

🔎💬 Inkeep AI search and chat service is syncing content for source 'Inkeep Agent Framework Docs'