UPDATE: This post is very old and no longer accurate
CREST API released by CCP Games in 2016 allows sending and receiving evemails in third party apps.
At some point in the past I took some time to test all EVE Online apps available for iOS and Android. Since the publication of that post, many players have asked me about an app that allows SENDING evemails, and not just reading them. I have bad news for you: due to the limitations of the current EVE API, such app does not exist. But I also have some good news for you. It does not exist yet.
With Retribution, CCP is planning to release a new API (called CREST), which will certainly allow you to manage Contats and is very likely to allow sending EVE mails, too.
For the third party developers, the most valuable resource is the current documentation preview, which has already been made available.
One more thing
CREST is an acronym of Carbon REST. The Carbon technology is the new CCP framework that has been in use since Incarna, and REST means it’s a RESTful protocol (wiki). It will be released in Retribution expansion, so one might say it is a “Crest of Retribution“. Now this is what I found when googling it: http://us.battle.net/wow/en/item/13375. Coincidence? 😀