Do you have 3rd-party cookies blocked? I had a similar problem in my own game, and maybe it's the same thing:
"if you block third-party cookies in Chrome, HTML5 games hosted on itch.io are denied access to LocalStorage and SessionStorage, even though it still works if you're testing using localhost, or on your own website." [dev log]
Other player workarounds: Unblock 3p cookies, or play the game in the itch.io app.
As a developer, I had to work around it by wrapping localStorage and sessionStorage access in try/catch blocks. I don't know if there's another way to detect that condition.