How to apply by application to a new team?

If you are in a club, you have to leave them first. Then when you look at the club's description, you'll see the button to "apply". Once you tap it, it will be up to the president or officer to accept or not. They can see your name and your level, that's it. There is no other communication, unless you discuss outside of the game somehow. And if they don't check their inbox often, them may not even know you applied.