I guess the League could give her the stone in return for a certain amount of time in their service, but that would make the League "mean" and manipulative which I think Riot has stated they were trying to get away from. Really, I think that's the explanation that makes the most sense.
Nami approached the League and said "Help me find this!" They hear her out, become aware of her impressive abilities, and say "Well well, we'll help you with that exchange, but on one condition..." Nami saves her people thanks to the League, but now she's bound in service for a set amount of years or until she's paid off her debt to them. It's the end of her bio's main story, but it doesn't paint her as being irresponsible or incompetent and there's still room to develop her afterward. She might try to seek out whoever it was that once delivered the moonstones in an attempt to ensure that a Tidecaller is never met with an empty beach again, for instance, or even make a pact with a new faction that can get the job done.