The only way would be to create a temporary email address specifically for that purpose and post it on here FAO the person. Not sure if the Site Rules allow that, though.
It's advised against but is allowed. Did it myself some time ago and when the address was acknowledged I reported myself for an act of sheer stupidity and the post was removed. Once I was contacted a new email was created and the
"posted" one canned.
Some users have private chats using very old AB threads that they know others are unlikely to look in on but for that to work you'd have to agree which threads to use. The best way is to create a temporary disposable email address which you post on their thread and then you take it from there.