Your first step should be to complain.
I'm shocked there are still businesses pushing things that require a set browser. It's reflective of a development process that went out the window over a decade ago. This is their active choice. That's the why.
You can usually work around simple blocks with User Agent Switcher. I've used it for some banking in the past. Firefox will just "pretend" to be IE and this is enough to get past most issues.
You might find that some things don't work. They might be coded so single-mindedly for IE that Firefox just refuses to continue... Or they might be doing capability-based testing to work out if you're using IE (which User Agent Switcher won't cover). If that's the case, your only good option is virtualising.
And by that I mean installing VirtualBox or VMWare Workstation, and installing Windows inside that. You'll then be able to run a real copy of WIndows and a real copy of Internet Explorer within your Ubuntu session. This obviously nukes a portion of disk space and takes time to boot but it's feature complete. You're not running into winegecko issues every other pageload.