is it possible to save your progress on the web versio
Runs in progress cannot be saved in either version.
Unlocks, stats, saved run variants and preferences should persist on the web version, in the form of a cookie stored on your machine. However, every web browser handles cookies differently. I have heard of people's cookies lasting upwards of a month, and others get deleted as soon as they close the tab.
The Windows version stores these files in C:\Users\UserName\AppData\LocalLow\SilverDubloons\Scrongly