Can I backup pictures that I have stored in a photo vault app?

It would have to be a menu option within the App. Or ask on the App developers support site. Each App independently controls where it allows you to store files online - each App has to independently support a particular cloud storage solution, whether it is access within the App to iCloud, or Google Drive, or DropBox, or Box or whatever. There needs to be a menu option to "save as" or something like that with the option of putting files into the online storage you wish to use.

If the developer of that App has built in iCloud storage access for them yes. But it has to be a feature of the App itself. Does it offer any access to online storage services?