The Community will be temporarily unavailable starting on Friday February 28. We’ll be back as soon as we can! To learn more, check out our Announcements blog post.
This is now possible!
Read about it here: https://airtable.com/api/meta
Simply make a GET request with the auth header, and see the tables listed:
https://api.airtable.com/v0/meta/bases/$BaseId/tables