The browser build does seem to have some issues which is really unfortunate, your description sounds super interesting. If you're able, I'd highly encourage you to open a support ticket on the Discord and see if you can re-submit a working copy, I really wanna see it!
Anyways, congrats on your first Godot game and first Godot Wild Jam entry! Exports are tricky when deadlines are this short, I also had to resubmit due to some issues.