Having the same issue. Also I noticed captchas are popping up from Cloudflare based CDNs, including Riot's own site.
My guess is that the League client uses a Cloudflare based CDN to handle user data, and Cloudflare at some point overnight fucked up and blacklisted a ton of IP addresses on accident. Noticed I'm getting 403s returned from the majority of CDN based GET requests which supports my theory since 403s signify unauthorized requests usually from blocked/forbidden IPs.
This doesn't seem to be an issue Riot can fix, but rather it is an issue on the end of whichever CDN they use (probably Cloudflare/AWS).
edit: Generally Cloudflare blocks IPs if they were associated with DDoSing. I have a feeling that overnight a massive DDoS attack occurred where thousands of IP addresses were spoofed during the attack, resulting in them being blacklisted this morning.