I can confirm that I can’t X Delete an item from the cart, but I don’t see any indicators that Cloudflare is caching anything that would cause this. It could be Rocket Loader. Try disabling that in Speed -> Optimization in the Cloudflare dashboard.
Then it’s probably unrelated to Cloudflare. Try using the “Pause Cloudflare on Site” option from the Overview page, then wait five minutes before testing your site.