most reach-in coolers have a simple door switch for the lighting. But on some makes they use the controller to both turn on the lights and stop the fans. I don’t have a wiring schematic for your box, so I don’t know if they use the controller, but because it has a door alarm and indicator on the controller, I expect it runs through it. So I would thing you would need to measure the switch continuity at the controller block. I expect you’ll find a loose connection there. Often with control being electronic, it may only be 5 volt sensing. Norlake uses a private controller, not like most other manufacturers.
I recommend you find the wiring diagram attached to the machine to verify. It may be glued to the back or in a bag attached in the compressor/electrical area.