According to the StackExchange Chat FAQ:
If you need to invite someone to your room, do it from the user's chat profile page or from the user card that pops up when you click on their avatar in a chat room.
Is this information correct?
Maybe I'm missing something, but I don't see a way to view a user's "chat profile page" from their SE profile.
Also, if you want to invite someone into a chat room, they are not already in the chat room. Hence they have no avatar in the chat room to which you want to invite them. That makes it challenging to click on their chat avatar.