Fuse: Check if the fuse that controls the license plate bulbs is blown.
Bulb socket: Ensure that the bulb socket is making good contact with the bulb terminal. Cleaning the bulb socket and contacts using an electronic contact cleaner, and then inspecting it to make sure it's firmly secured in place.
Light switch: Turn the headlights on and see if the dashboard switch light works. If it isn't lighting up that fuse controls everything, check that fuse.
Wiring: Examine the wiring to the license plate lights for any visible signs of breakage or corrosion. Repair damage.