You don't need an API to imitate Imgur/Reddit and link to Mobafire, or Lolking or any other site that has done the work of aggregating the stats of people you want to shame. You can also link to League's site and League's reddit, but neither of those would matter for these purposes.
You don't need an API to toss up some forum software, restrict or ban doxxing and start monetizing it the way Twitter does.
You /may/ need an API to be able to archive posts, but not Riot's API. Generally speaking in order for the site to survive, you would need to delete posts after anywhere from 24 hours to 2 weeks unless they are stickied. You would need links to Pastebin for Riot support tickets and other emails ofc.
No one going to such a site expects the kind of staff or uptime that even 4chan managed, either. But it's not hard to start selecting specific players to shame and then pasting tidbits from the posts on Twitter either.
There's plenty of ways to monetize it without pissing people off or inadvertently giving them malware. It wouldn't be easy, but it wouldn't be the most difficult thing in the world to monetize for a few years either.
In what world would you need Riot's permission or API to create a 'some men want to watch the world burn' environment? You are at least as exposed as the people trying to name/shame trolls because guess what, they have to confirm with matchmaking information and screenshots. The board culture simply entails the opposite of Riot's, if you censor the identifying information, you get downvoted or banned.
It's really not all that different from Facepunch forums banning for poor grammar. You are simply facilitating the mob mentality and forcing Riot to act quicker to implement votekick, match forgiveness for AFK's and 3-2 surrender votes. Circumventing the policy of their support re: getting news about the names of the people you got punished isn't punishable by any rules Riot does, nor are you doing something as stupid as providing a drophack.
All you are doing is providing a service Riot cannot for PR reasons, forcing the community to police itself and circumventing the atrocious and unworkable policy of player behavior modification in favor for ostracizing them.
You may have to downvote/ban for not accompanying your whiny rants with screenshots and match history links, but none of these involve implementing an API whatsoever. You /will/ have to be competent enough with site security not to allow Lizard Squad to hijack your site with outdated software given the nature of your site, but if you operate the site such that this is an eventuality instead of a preventable outcome, it's easy enough to mitigate the damage via backups or abandon the site entirely when it happens.
There is literally nothing any MOBA developer can do, to slap you with a cease and desist for running an imitation of the Better Business Bureau for medium-profile MOBA players.