Wait... so... why is she in the League? Her quest is to find the moonstone, right? How would being in the League help her accomplish that?

Publicity, probably. If she can garner sympathy from the public for her quest, they might be inclined and able to help her find that moonstone.

Perhaps also so that the summoners would allow her to access their libraries or whatever they might have, or to rub shoulders with important people who could also help her out.

I considered that but that answer seems kind of generic, you know?

"Hey, we have a champion who doesn't fit in."
"That's alright. Have them in the League to garner publicity."
"Oh hey, that works!"

You see what I mean?

We have about three to five of those in a row for the last few new champions in case you hadn't noticed.

On the upside, new skins are becoming downright fricken amazing.

maybe she plans to loot the corpses of the other champions.

She doesn't know much about the surface world, just doing random things looking for this moon stone.