Well, as long as you play to win and play fairly well, even with your impairment, you should be fine. Even if League is kind of one of those games that requires two hands to play (between mouse functions and ability hotkeys), I'd imagine you could probably still do it.
I think, though (and I'm definitely not certain if Riot is okay with it), but it could also be possible to get a special mouse to map ability hotkeys to? You'd probably have to ask about that, since I do know there are gaming mice with buttons for special hotkey/macro setups, but, again, no clue how Riot views those. So, definitely ask about that possibility if you can.
And, if nothing else, take inspiration from Halfcoordinated - a legendary speedrunner who plays one-handedly. (Granted, he doesn't play League as far as I know, but his ability to play videogames is absolutely amazing regardless.) If he can play NieR: Automata or Momodora: Reverie Under the Moonlight with only one hand, I'm certain you could play League with one.