I've had this problem ever since I started playing league about a year ago, and from all of the tests I've ran I've come to find out that like a lot of you have found out that it isn't your firewall on your computer that is the problem, but it also isn't riots fault, it is actually your internet service providers fault, for me it started doing this after a patch a long time ago, for some it's been more recent, buy through talking to my ISP it's because something new in the game patch has caused your ISP to question the server and block it, usually automatically, I know it's the ISP because I've tested it with many different ISP's at friends and relatives houses and some would block it completely, some would block it at first but then would let it through after clicking retry a couple times, and then some it would always go right through no problem. Unfortunately my ISP tech support is to incompetent to fix it when I called them but I've had friends who have, my ISP did say though that if Riot called them and talked to them then they probably could figure out why they're getting their service blocked.
TLDR: it's not riots fault it's your Internet service provider, but if contacted may be fixable, if not Riot themselves would have to contact them.