Since you don't want to mess with just any emulator that "emulates" the Android platform on your PC, there is this one that I highly recommend...
BlueStacks
Of course, the answer to all your problems is the #1 Android Emulator - BlueStacks. BlueStacks is the Android Platform Emulator, which means that it kind of emulate the Android OS, but only for the gaming (app) part. And yes, you can download and play the most mobile apps and games that exist at the Google Play Store.
How-to
As I mentioned above, it's very simple; firstly - download and install BlueStacks Android Emulator, then create a Google account if you don't have one already. Use your Google account with Google Play Store in BlueStacks, and you're all set to download the one and only - Fallout Shelter!
Links (needed in order to play Fallout Shelter on PC and MAC):
Check out Fallout 4 Collectibles and Other Cool Deals from Amazon, and also our Fallout Shelter Portal and have fun!