If you have already bought the game, it is linked to your itch.io account. The app should download the updates for you. If that's not working, you can just reinstall the game without any trouble.
Just in case you are using your browser to download games, you have to log in with you itch.io account and you can redownload the game again.
If you have bought the game without an account, check out this link: https://itch.io/docs/buying/already-bought