The game does work for me on Firefox, it just opens in its own window, although it might be still causing bugs since I've never actually seen the sword parry anything, but that could also be an issue in general. However, I can't switch to another platform since I've already made quite a bit of progress and there doesn't seem to be a way to export/import data that I can find.
When i try on my own firefox or any other computer at home, even at friends, no one can play on firefox. it opens a new window like you said but it becomes an error 404 with a skeleton image in the background.
If you are aware of how cookies work you can go in to your firefox cookies and find Knota. There should be a "your-character-name.save"-file. Move this to your other platform's Knota Cookies.