Hi Lisa...welcome to the Forum
I think the easiest and cheapest method to track down the fault would be to use the 'wiggle test' which involves the ignition being switched on (engine off) and reverse gear being selected. Get someone to sit in the drivers seat to report back on whether an image shows.
Next, 'wiggle' the connections on the tailgate wiring (put tailgate up and down slowly), connection on the gearbox for reverse gear switch, any other wiring that you can find associated with the camera. Also lightly tap the camera housing to see if this gets it working.
Usually, if there is a poor connection or wire that is breaking up in the tailgate wiring where it articulates when open and closed, etc. then this could be the cause of failure.
If this initial diagnosis doesn't work, then an inspection of fuses, etc and failing that, take it to the dealer for a diagnostic scan of the system.
Let us know how you get on with it