@David If you can find the chat user (which is not trivial if you don't already have the user in a chat), you can visit their chat profile and there's a link there. For example, if I click on your name in this room (or if you click on mine), I am taken to chat.stackexchange.com/users/485972/david which has this button:

Do they get some kind of notice of the invite?

Yes, they get an inbox notification

The best way of finding the chat user is going to chat.stackexchange.com/users and searching there. However, only users who have visited chat have a chat account. Many of the users active on the main sites have never used chat, so won't have chat accounts (they're automatically created on the first visit).

ok thanks.