Fix For League Client Not Opening
Basically, your antivirus program is removing league files thinking it is a virus. To fix this you want to regain or restore those files. I'm not too sure on other antivirus programs but on avast if you go to scan > scan for viruses > at the bottom there should be an option called "Quarantine (Virus Chest)". Files from league will be located there, so select all of them and restore and add to exclusions.