I had a play with installing the RoundCube webmail system on my server yesterday and was pretty impressed. It does seem to do what it says on the tin and act as a AJAX-y web front end to an IMAP server.
I failed to find such a (public) API. Do please link in comments if my Googling was insufficient.
Then I stumbled upon a description of the Maildir on-disk mailbox format which my IMAP server was getting all the mail from and it struck me that, actually, the Maildir format itself is RESTful. It certainly ticks many REST boxes:
- Authentication is devolved to some higher power. In REST APIs this is generally some HMAC sent with the request or using the Auth mechanism built into HTTP. With Maildir, any authentication/permissions/etc are handled by the filesystem/OS.
- Data are organised as resources. Each email is a file with an unique ID located by a semantically valid resource locater (i.e. a folder subdirectory in Maildir++ boxes).
- It is stateless. Maildirs were designed to allow stateless, lock-free concurrent access. Just what a RESTful API craves.
 A funky little beret naturally!
 Well, locking handled by the file system anyhow. It has been pointed out that there are some conditions where Maildirs are not the panacea they appear in this regard but for the purposes of this discussion I’ll stick my fingers in my ears about it.
 Reading. i.e. without the actual sending mail part.
Update: So it appears that one of the protocols Exchange and Outlook speak for exchanging emails is WebDAV. Great minds…