Just wondering if its even possible, it would be fun to be able to run this in the background while I'm working on the computer

If only, I'd love to play on a bigger screen without buying an iPad. But I don't see this happening, but then again, I'm not in the loop when it comes to things like this (or it's legality for that matter)

To the best of my knowledge there are no IOS emulators. There are a few simulators out there and one or two of them bill themselves as emulators. The short answer is currently there is no way to run Fallout Shelter on a windows desktop.

There are some Android emulators so it may be possible to play Fallout Shelter on a windows desktop when the Android version is released, but...

Emulation software is rarely 100% perfect. There are always some things that won't work and other things that work but not very well. So maybe later you will be able to run Fallout Shelter on an emulated Android device on your desktop, but then again, maybe you won't be able to.

I'd just like a version of it that works at all now. my iPad version has stopped functioning for any vault that exceeds 27 members. literally the moment a 28 dweller inborn it crashes never to display again, the fun thing it keeps sending me popup notices as to whats going on, even if i can't play

Well, it is obvious that it is still running in background, otherwise you wouldn't be getting those popups. Have you tried fully stopping each instance and restarting?

What version of ipad are you using? I had an early iPad Mini and after 30+ dwellers the app just crashed constantly. I read that the older iPhones and iPads don't have enough RAM.

You can emulate a Mac on Virtualbox, install Xcode there and use that to emulate an iPhone. However, you won't get past the splash screen because you'll be running it on x86 architecture. I tried getting it to run (there are ways to translate to other architecture), but basically it's too much work even if I get it running for others to even replicate this. You'll lose an entire evening setting up the mac emulation. It's better to wait the remaining 6 weeks for an easy install on android, which is easily emulated on desktop because Android isn't created by greedy people who want people to buy their overpriced stuff.