Interesting. Progress should actually be saved automatically, but it looks like that isn't working on the itch.io site for some reason, probably something to do with how itch.io embeds/sandboxes the game.
I'll have to look into this, but in the meantime if you play the game at https://cocoamoss.com/nyamos-adventure/nyamos-adventure.html or https://ddrkirby.com/games/nyamos-adventure/nyamos-adventure.html the auto-save functionality should work.
Edit: So, this is due to the way that chrome/safari/etc block cross-domain access to indexeddb in conjunction with how itch.io hosts games in an iframe. I'd need to implement an alternative saving mechanism in order to get this working on itch.io.
In the meantime there are two workarounds:
1) You can enable third-party cookies (in chrome this is called "allow all cookies".
2) You can play the game on cocoamoss.com or ddrkirby.com.