Although @sanity31415 has offered to get in touch with his 33mail product, there's also the option of standing up a REST API server in front of your mailboxes.

There is an open source product called IMAP API which is available under 2 licenses depending on your needs: AGPL & MIT. It recently had its own Show HN [2].

[1] https://github.com/andris9/imapapi

[2] https://news.ycombinator.com/item?id=25372987