Hello Jason,
From what I can see from your post, the camera is not working on Facebook Live but is working normally in the OS, correct? Are you using an app or logging into the website with a browser, like Safari?
First, try updating the OS to Catalina 10.15.4: macOS Catalina 10.15.4 Release Notes
Second, go to the App Store to verify the app is up-to-date (if applicable): App Store > Updates > Update App
Third, Check in System Preferences > Security & Privacy > Privacy > Camera > Allow the App you are requesting.
Hopefully this will help!
Cheers!