Is it possible to build a SMTP/IMAP client that can run in the browser that uses only Javascript?
It's currently not possible to do in a normal web app in the browser. It is however possible to do in packaged apps with access to privileged apis:
If you're looking for a battle proven library, whiteout.io develops and maintains a well tested IMAP / SMTP / MIME-codec implementation in pure JS:
It's MIT licensed and free to use.