I have done some testing, I don't know why the HTML export in Godot is broken for this game. I have checked the standalone on Linux and Windows and those seem to be OK. I apologise for the inconvenience.
Edit: I figured out it's one of the plugins I was using, it's fixed but unfortunately not in the public release :(. I was able to patch it by hand and export again, and now it works.
I can't update it since the jam is closed. This might be against the rules, but in any case, I would rather have people being able to play the game and have my entry not count for the competition if it breaks some rule I couldn't find, so here's an alternate link:
https://4kxz.itch.io/winter-coldfix
It's the exact same version I uploaded for the jam, identical to the Windows and Linux versions, with the latest branch of this plugin applied.