To get this operational, do you want the original cooling equipment, or do you just want a functioning built-in cold box?
I ask because if you are open to replacing the components, you should be able to put a modern compressor, evaporator, condenser and new lines in. You'd have to measure out the parts you need and depending how easy the lines are to access, it'd be simple to convert this to a modern setup. By modern setup, I mean self-defrost, air-flow control etc.
My Great-Uncle owns a refrigeration/welding shop. Growing up and still to this day, my grandparents have always had a walk-in cooler in their basement with remote refrigeration. I've been drawing and toying with the idea of building base and upper cabinets as homemade refrigeration and have been gathering parts from the shop to start the process. I've always loved the idea of something completely custom to a space and .. not just a box brought home to fill an opening.
I am so thankful pictures were posted, I am having a hard time trying to get people to imagine what I'm trying to do. If I may ask, what is the construction of this unit? Is it an insulated wood cabinet? How thick are the doors?