If you have a clear cookie/cache option on exit in your browser that might be the cause of the issue.
Atm the save system is using the browsers local storage.
Do any other (Unity) games load? If not, my first thought would be that you have a strict cookie/JavaScript blocker in your browser. For example: the browser 'Brave' has their Shield feature which causes this problem.
It seems to be a problem with blocking third party cookies.
In Chrome you can go to this settings page to enable third party cookies if you so desire. (Or if you have an extension to handle the blocking for you, try disabling that)
In the Brave browser, simply disable the shield feature by clicking the lionhead at the end of the URL bar and toggling it off. (Remember to toggle it back on when you're done playing).
This has fixed the issue in the past, so I hope it works for you as well.
If this does not solve the issue I'll have to investigate a bit further and I'd like to know which browser you're using.