The API does not provide any method to determine a user's email address and it is unlikely to do so in the future.
This would be a risk to both the user's privacy and his/her trust in Stack Exchange. And, a pressing need for this information has yet to be demonstrated, regarding the API.
Note that Stack Exchange refuses to even provide the user's email hash via the API, even though this information was once available via the data explorer.
(Providing email would be much more egregious than providing the hash.)
Even a typical auth warning like:
"This application asks for your email and will do unspeakable things with it." is not sufficient protection.
A naive/underage/frazzled/rushed user often won't realize the risk/ramifications when they see such a notice, before clicking "approve". But they will remember that "that sucky Overflow site gave away my email!" ;-)
If your app really needs the user's email, ask the user for it directly.
They then can decide if they want to divulge that information.