I believe I accidentally uninstalled the camera software on my Mac, can I reinstall?

The fix will be to reinstall macOS from Recovery mode.

Please follow the steps in this support doc for the Intel Processor Mac.

How to reinstall macOS - Apple Support

Reinstalling the OS does not have to include wiping the drive or erasing anything. It simply lays a fresh copy of the OS on top of the existing installation, replacing damaged and missing components - your camera software, for example.

If you find yourself attempting to erase or format a drive or volume - don't. Quit the process and start over.

Thank you for clarifying that. Which specific app are we talking about? Most of the apps which are installed with the Mac's operating system cannot be removed. Have you checked in the Applications folder to see if it is still there? Otherwise, you should be able to find it and download it from the Apple app store.

The same way you installed it the first time, either from a CD or by downloading the app from the camera vendor's web site.

If PhotoBooth is unable to access the camera, then most likely the camera has a hardware failure since the green light is on indicating that the camera is physically seen by the system and the app is attempting to operate the camera. You will likely need to have Apple or an Apple Authorized Service Provider examine the Mac to confirm & provide you with a repair estimate, otherwise you can try using an external USB camera.

Also, don't forget to check security preferences to make sure the Camera can be accessed by those third party apps (I don't believe this applies to the original Apple apps). If it only happens on a secondary macOS user account, then maybe the ScreenTime (formerly Parental Controls) preferences are preventing access (I don't use them so I'm not certain if it would apply).

Control access to the camera on Mac - Apple Support

Not sure about the file but if you reinstalled the OS all files should be there. So it could be camera mechanical failure.

Looks like you may be missing a driver. Reinstall the OS.

Not that I know of. Possible that the camera is bad. How have you checked camera function? To check camera function try to open photo Booth and see if that works.