Hey Harem!
Similar to the downloadable version: Achievements, fame and perks are saved between runs and gameplay sessions, you should also be able to close the game mid-run and come back later, HC2 automatically saves after every player action.
If your game is not saving correctly please let me know as this is probably a bug. But first, make sure you're not using incognito mode or any other stuff that might mess with or clear your browser's data.
There are also plans to implement a manual save in easy mode, but unfortunately I don't have a release date for that yet.