I have a 2017 iMac Pro running Monterey 12.5 . When I launch Studio within Canvas, I can access my computer's camera in Webcam Capture, but when I choose Screen Capture and want to record "Both" I click on the "Select Webcam" option and no camera list appears. The link just turns blue. Since I am unable to select a camera - it never asks for permission to use the camera - ergo, Screen Recorder does not appear in the apps list which can use the camera in Privacy Settings. If I boot in safe mode - I CAN choose a camera from a list, but my computer cam is not on that list. I have uninstalled, reinstalled, tried different browsers, trashed every app that I don't need along with their preferences. My IT folks are working with me, but so far no solution.
I tried the process using another iMac Pro in the department and it works fine.
Just hoping to clarify, are you trying to use your built in camera and not an external webcam? I wonder if other similar apps recognize your camera, such as screencast-o-matic or Facetime. Here is a page with a few options to try, such as checking the Screen Time settings and resetting the SMC.
The issue is with the built-in camera on my iMac. The camera works for every application - including the Webcam Capture option in Canvas Studio, but the Screen Capture option in Studio will not let me select a webcam.