You can google where the save path is....I just created a backup save folder and copy my vault saves into a new save folder using the regular file browser on my phone.
I have read too many disasters on these forums to not do this...
meh I dont trust cloud backup. So I manually backup to my backup HDD.
A copy/paste from one of my previous posts...
There is a way to backup your saves. Get into your file system (depending on your manufacturer you might need to get one from Google Play, like ES File Browser) and go to
/storage/sdcard/android/data/com.gamesas.falloutshelter/files/VaultX.sav
where the "X" in VaultX.sav is the number of the vault in question from top to bottom in your savegame list in-game (so 1, 2 or 3)
Copy and paste that daily/every other day to a safe location or email it for all I care, but that way you'll lose little progress and/or boxes.
Yeah I already figure it out but hope they make this post or some kind of post as sticky post, that way ppl don't have to look everywhere