Do they have a GitHub action to run in GitHub, similar to Claude?

Yes https://developers.openai.com/codex/cloud/code-review

It isn't quite the same. The Claude Code action can be easily integrated into a workflow to fire automatically (like when a PR is opened).

The Codex support at the moment requires adding a comment "@codex review" which then initiates a cloud based review.

You can, however, directly invoke Codex CLI from a GitHub workflow to do things like perform a code review.