Hi dpedersen439,
We want to help with why you can't save scanned documents, however, we need to isolate the issue, which is why we have asked you to test without an Apple app.
Did you have a chance to test with using Image Capture?
If so, what happened? Were you able to save?
Also, which app are you using, is it one from HP? If so, is it up to date?
Use the App Store to update apps on Mac - Apple Support
If up to date have you contacted HP to verify that their app is compatible with macOS Ventura?
Contact a third-party vendor - Apple Support
Thanks!