This is now possible by filling out a form to request elevated permissions:
- Go to https://support.twitter.com/forms/platform
- Select "I need access to special permissions"
- Enter Application Name and ID. These can be obtained via https://apps.twitter.com/ -- the application ID is the numeric part in the browser's address bar after you click your app.
- Permissions Request: "Email address"
- Submit & wait for response
- After your request is granted, an addition permission setting is added in your twitter app's "Permission" section. Go to "Additional Permissions" and just tick the checkbox for "Request email addresses from users".
Note that user e-mail address has to be verified, otherwise Twitter refuses to provide it. (See include_email parameter description in elevated permissions doc page.)