Have you had any issue saving progress in any other game? The save game files generated by QuestORama are a few kilobytes, so it seems HUGELY unlikely you're out of space. Are you low on space?
I would guess, for some crazy reason, your folder is file locked, so it doesn't allow saves. Have you updated your firmware recently? I tested on the release with hand tracking, so I'm certain it works with the latest Oculus firmware.
Just to be sure, try one of the older versions off GitHub directly, see if they work better: https://github.com/MaxSMoke777/QuestORama/releases
The save game system is new, so not sure if you'll see it fail even in the second to last version. Also, have you tried reinstalling, not just clearing the app data? *Sometimes* things go sideways on computer systems, it really could be just a bad install.