Does Itch.io have any rules around saving data in a player's browser? I'm making an HTML5 game and I want to save the user's progress in localStorage.

Just do it, like everyone else. These are not cookies, no data is collected on any server, so there are no privacy laws or other reasons not to do it. Except that you're using a tiny bit of memory on the player's computer (I believe 5MB), but no one will mind. If you encode your data well, you'll probably not even get close to 5MB.

EDIT: Actually, if you're sneaky about it, you could read data from local storage, and based on what you find there (like high scores, or time played), send information to a server, still tracking the user that way. That might be against strict European privacy laws without user consent.  So don't do that, unless you're aware of all the implications and laws. But that's not what you were asking about - just storing progress locally is always okay.