I'm having the same issue also. It's really frustrating.
Same issue on Android. Updated settings to allow app cloud backup. Tried copying game files, uninstalling and reinstalling the game. No luck. Help please. I'm hesitant to continue to play and grow my vault without knowing whether or not it would be recoverable.
I play on iOS so I can't really help with Android but in iOS you have to have iCloud on and you have to activate iCloud Drive within your iCloud settings. The app saves a file to your iCloud Drive that backs up your vault. It can also read from this file so if you use multiple devices and they can all access your drive, then you can open your saved vault from multiple devices. Play for a bit on your phone and then switch to an iPad to continue if you like for example. I don't know how Android does it but I would guess it perhaps works in a similar manner using Google Drive?
First off, you need an iCloud account to be setup on your device. Then under Settings --> iCloud, you should see a setting for iCloud Drive. Turn this On. Then go into the game and bring up your vault list. Tap the little cloud icon next to the vault and it should then back up your vault and not give an error message. If everything went right, you can go into Settings --> iCloud --> Storage --> Manage Storage. You should see a Documents & Data heading there and under it will be a listing for Fallout Shelter. iCloud can be set to back up your game but this is strictly for restore purposes if you change devices or have to wipe and restore your device. You'd lose any progress since the last time your backup to iCloud ran though. With the drive, the game itself will keep your vault data backed up and current to whenever you last played.