I'm a moderator on a sister site (TeX.SX) and so far I know there is no method to contact a user directly for normal reasons. The moderator tools allow you contact users privately for serous reasons only (and this seems to now automatically suspend the user!). Using the moderator power to reveal the email address would be one way, but some users might not react well to this.
One way we do it sometimes on TeX.SX is that you add a comment to some post that user posted or commented (incl.
@username of course) so he gets notified. Then afterwards the comment can be removed again. It's only semi-private ;-)
In theory you could also invite the user to a new dedicated chat-room you can create, but all conversation will be public and stay around forever, so this is hardly what you want.
Finally, you could go to the moderator chat channel and ask more experienced moderators how they handle this. You find a link to it on the moderator dashboard page in the right sidebar. Asking your question on https://meta.stackoverflow.com/ might also be useful. It's not a homebrew.sx specific one after all.