OK, I found the bug. Internet Explorer should work fine now.
Here's a tip, you can try "Safe Mode" anytime there's an issue. It disables any customizations or plugins I've put on the site. Knowing that it works in Safe Mode makes tracking down the bug a lot easier.
EDIT: apparently the live chat doesn't work at all on IE. I have no way to fix that one. I still recommend people switch to Chrome/Firefox/Edge.