If you have upgraded to OS X 10.14.1 or newer and are using LockDown Browser for the first time, you will likely need to manually grant LockDown Browser permission to access your camera and microphone.
To do this, on your Mac go to: System Preferences > Security and Privacy > Privacy
Click the lock icon in the lower left, then enter your admin password. Afterwards, click on "Camera" in the left hand menu and then click the check box next to LockDown Browser, as shown in this photo:
Perform this same action for your microphone:
Afterwards, click the lock icon to prevent further changes.