As far as I know, no. Firefox is my default navigator and I don't have any issues, neither with the saved games, or the gallery.
There was an old bug, with one of the updates (0.20) where I named it differently by mistake (index.html instead of the correct name Index.html) If you had a save with a different named file, and you try to load up another, it will say about the Web Storage API. There are two ways to fix it: One is clearing cache/cookies of the browser which is... not ideal)
The other one, is to load the old html file (if you revert back to the last version that worked for you you should be able to load the game). When you're inside the game, erase EVERY SAVE on the list (you can load up your most recent, and save him to a external file. That's fine, you can load it up later) and then coming back to the latest version, and there you have it.
Another easier solution would be to switch to Chrome, Edge or Opera.