If you were able to click on buttons in game, so you must be able to move items too. I don't know why doesn't it work, but try to use other browser instead of FF, maybe it should work (if it isn't still, so the problem could be because of your OS)
I finally figured out how to move items: it's not by mouse cursor, but by dot in the middle of the screen T_T. Still no buttons though. My only problem now is camera control, which requires to Alt-Tab in and out of browser window.
I'm also getting this is browser console:
Failed to create agent because there is no valid NavMesh Lewd Cashier HTML.framework.js.gz:10:34712
Request for pointer lock was denied because Pointer Lock API is disabled by user preference. Lewd Cashier HTML.framework.js.gz:10:181325