They may have spent a hundred grand like my employer for a browser based
payroll/accounting/student registration/records package. They don’t want it broken by a browser update. Nobody gets paid if the payroll software doesn’t work.
Why in the world such expensive software relies on something as unstable and constantly changing as a browser is beyond me.
In theory, because it’s platform independent - so you don’t need to worry when you upgrade your desktop OS, or switch to a different desktop solution (thin client, or Linux, or whatever).
In practice, any significant change to anything can cause problems and needs a careful approach - and the more of these things overlapping that you have to juggle at the same time, the harder it gets to achieve anything with any of them.
This is exactly what I do for a living - I have to manage roadmaps for 100+ applications/systems, against a bunch of other roadmaps for server/desktop/browser/database architecture, and legislative and business requirements. From the customer perspective, it very often looks like nothing is happening, but that’s because it’s a hell of a complex task.
And if that programmer is long gone? As others have suggested, the web app may be a black box that is to all intents and purposes not fixable because no one available understands the guts of it.
Browsers (as a client platform for web application development) have been stable and reliable for a good 5 years or so now. Anything built to modern standards is both more stable and easier to maintain than antique Netscape/IE6-era junk. Cross-browser compatibility is basically a solved problem with modern browsers and standards, and even mobile compatibility is pretty straightforward.
Unless you built it all wrong from the ground up, and have spent years failing to dig yourself out of the pit you created. In which case you blame everything on browser incompatibilities, and insist users stick with decades-old versions of IE.
Applications that do their job just fine generally do not get upgraded just because something better comes along. If it did, it would have been a lot sooner that a couple years ago that various banking systems finally migrated off of OS/2, and COBOL would be long gone.
So programs that require IE6 to function continue to exist until there is an actual functionality reason to change the program significantly, at which point it will probably be rewritten to be a little more browser-independent (hopefully).
It is unfortunate that a lot of these programs were written when the “Nobody ever got fired for choosing Microsoft” attitude was such that everyone just assumed that all of the proprietary bits and pieces IE supported at the time would continue to be supported moving forward.
No. Not really. The allowed alternatives are Firefox and Chrome. IT guys tend to be humor deprived and take it where we can get it. The reason for humor deprivation tends to be the lack of an audience that doesn’t need it explained. And here I have a captive audience. So sorry… 
Shouldn’t be. It is. Welcome to the real world where not all apps are developed and maintained in house. And even the stuff from in house that still works under IE7 takes a back seat to apps that don’t work or that need to be finished for current projects.