Samsung S22 cell phone is regularly making the camera clicks

Have you tried uninstalling the ring app and see if no longer getting the sound. You can always reinstall if after a short test period.

It's possible for hackers to activate a phone's camera without any indication on user end. If the issue persist, you may want to try a couple of other free security scanning apps. My thoughts are it's most likely notification sound for some particular app. Despite having disabled notifications.