Is it possible to transfer a game to another device?

If you have a look at your vault list, there is an option to turn on "cloud save"

If you turn this on, the next time you log in to your game center or iTunes account, you can download the games onto your new device.

The vaults are .sav files on your phone. I have an Android N device and the save file is located under /Internal Storage/Android/data/com.bethsoft.falloutshelter/files/, the names are Vault1.sav and so on (obviously, the path will be different for your iPhone, you'll have to figure that one out yourself or google for it). You can move these files between devices and that way, you can also transfer your vaults. This should work without problems between iPhone and iPad, not sure about the Mac tho.

I did the same a few years ago when switching to a new smartphone and it worked perfectly fine.