Make a profile field for that but don't make it compulsory (not everyone will appreciate their emailaddress being visible to the public)
This would do too, is this an extension?
No, it's default feature (but it's subject to permissions settings).
Under email settings in the ACP you want this
to be set to "yes".Users send email via board:
Instead of showing the users email address users are able to send emails via the board.