There are a couple ways to invite people. One is in-game by clicking on a persons name and choosing "Invite to Clan". This feature is only available if you are a Commander or Lt. Commander. Another option is in the mobile app. They can search for your name and click on the "Invite to Clan" button.
You can also search for the clan and click "Apply" and then they can try to accept it.
If they send you an invite it should show up in the app or in the game. You should be able to accept it either spot.
If the problems persist, I'd recommend leaving your clan altogether (see #3 and #4 in this thread if you are the clan leader) and then trying to get the invite.