First, don't post the image publicly, as it's against the TOS to name-and-shame someone. Secondly, I definitely would have reported them at the end of the game for being racist in the pre-game lobby. Just use the text box to explain the situation in the report.
Failing that, you can send a support ticket to Riot with the screenshot included if you feel it necessary. You will get a copy/pasted response saying that they appreciate the ticket but can't tell you if/when they punish the player, and that's mostly a privacy thing Riot respects. After that, your job is done and you might as well delete the image and forget about them.