Wow, it's worse than I thought. At this point I'm pretty sure that it's not just different browsers that're the issue, it's a combination of different browsers and different system display gammas (!?).
The way I'm defining clickable areas involves rendering images with big blocks of colour (blue for one item, green for another, etc.) on a canvas, and then checking the pixel colour at the location that was clicked. But some sort of arcane gamma correction details in the PNG standard can apparently cause PNGs to render slightly differently on different systems (see https://hsivonen.fi/png-gamma/ , I think). Great! You love to see it. And in this case, specifically the areas that I coloured green are broken for some and not others.
Thanks for the help! Hopefully I can fix this when I have time & get permission to upload the fixed version before the jam ends...
Update: I fixed it, I think!