In Settings -> Saving you can export your files if you want. I wasn't aware the Web build had so many crashes, though.
Okay, so I can export the files, but there's no way to import them in the web build. I had hoped that maybe initiating an export would also persist my data in the normal way. But no, that doesn't seem to happen. And as far as I can tell, there's no way to even know when a save has happened so I can close the browser shortly after.
This seems like a bad situation for players on the web.