I have the same issue, started today. My internet seems to be working fine - except chrome seems to be having issues doing google searches and loading gmail. Chrome is my default, but I read that League runs Internet Explorer, which can reach those sites without a problem, so I'm not sure if Chrome is the issue. In addition, I have friends from out of state that run chrome as their default on a mac, and they've had no problems. If geography is a factor, I live in the West and they are in the southeast.
I uninstalled and reinstalled league to see if it could be fixed that way. No luck.
I can't get in to ANY games. I've been trying with customs so I don't get punished but it is incredibly frustrating.
After about 30 seconds of the black screen, I see a white box that says "Unable to connect to the server. Please check your network connection and attempt to reconnect to your game." My options are "retry" or "cancel". Retry gives me 30 seconds of the black screen before the error message again. Cancel takes me out of full screen and I see "The game has crashed. Please reconnect"
My default browser is chrome, i have internet explorer installed. I am using a dell latitude running windows 10. Everything is updated to the best of my knowledge.