Can't login on more than one computer

Hi I'm not sure if this is the right section to post this but I've tried everything including emailing riot. Our school is trying to start a league club and we are at the stage of making sure the game works on every computer. But the problem is that we can't login properly on more than one computer. Normally when one is logged into it will be fine. 2 will sometimes work. But 3 at the same time and the login will send the other 2 into Trying to Reconnect. I'm not sure how to fix this, my teacher has been on the phone with riot support and they couldn't help, I've emailed them and they couldn't help. Does anyone have any suggestions?