Only Leaders, Co-leaders, and Elders can invite other players to co-ops. No other co-op members can do so.

You can send a co-op invitation to any of your in-game friends. Just go to their towns and tap your co-op badge in the bottom left-hand corner. The invitation will be automatically sent to your friend’s Co-op Center Inbox, at which point all you have to do is wait for them to accept it!