I'm not sure, but I think web browser are limited to talking http over the network, so you would need a proxy some where to allow for this -- architecturally similar to eg:

  http://engineering.linkedin.com/javascript/vncjs-how-build-javascript-vnc-client-24-hour-hackday

I've thought a bit about writing an email client (for the console, really -- but also thought about what would work on an Android phone) -- and I think making your own protocol and a server that proxies that protocol to imap/pop would be the way to go.

As far as I can tell this is how gmail works, and I seem to recall the original author of the SUP email client went in a similar direction on a new project (but I can't find the link atm).

I really don't like the idea of a "fan in" structure, where a proxy will end up knowing all usernames/passwords for every user's every email account. It would be better to have the "app" speak imap/pop -- but as I understand it -- implementing a useful subset of IMAP is not a trivial and makes ponies cry.