Hold down the Option key and go to the ➙ System Information menu option.
One the System Information report papers go thru the Hardware section (USB if it's a USB webcam) to see if the camera is recognized.
As a first fix boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.
NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following;
• Verifies your startup disk and attempts to repair directory issues, if needed
• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)
• Prevents Startup Items and Login Items from opening automatically
• Disables user-installed fonts
• Deletes font caches, kernel cache, and other system cache files
NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.