Is it possible to do so?

Bumping, anyone know?

The iDevices use Apple's iCloud and the Android devices use Google's drive, so they can't do it automatically.

Does your iDevice have a way to browse the files on the device? With my Androids I just use the file manager app(every Android I've seen comes with one) to browse to Phone storage\Android\data\com.gamesas.falloutshelter\files\ and email myself all the vault.sav files. I've manually moved vaults between Androids this way. If you can poke around the iThing's file system, you could try doing it the same way.