Is there such a thing as a lightproof window AC unit, or has anyone come up with a simple workaround that does not involve modification to my apartment?
I feel your pain.
Here in Norte Tejas, not having an AC unit means not working in the darkroom 9-10 months a year! My first permanent space was in a rent house in the 1970s. My landlord had been a keen amateur and had built the room, but it had no heat or cooling. He graciously let me install a window unit at my own expense. I do not recall it leaking light. I did a similar thing in another house in the 80s and do not recall it leaking light either.
Then I had a nice darkroom (finally) with central heat and AC from the building. Oddly enough, I had to put a baffle over the register vent to keep light out from the ducts, but that was successful.
My current darkroom, built out in 2010, has a window unit in the wall. It was there when I bought the house and it leaked some light. I replaced it a couple of years ago, and the new unit glows, too. So, either I did not see the light leaks 30-40 years ago (entirely possible), or the older units were built differently than current ones.
But, to answer your first question: probably not. The second question: yes, you can pretty much light proof one.
Without blocking air flow, you will need to get foam weather stripping all around the case on the inside. If you can take the front off - often necessary to replace or clean the filter - it should be obvious where you can (and cannot) do this. It will also be obvious where it is needed, if you let your eyes adjust (several minutes, as has already been suggested) and get a good look at the unit in the dark.
It may also be possible to put some dark filter material in front of the air output grid, which will cut down the light considerably. Just be careful not to cut down the air flow considerably, or it could create problems and damage the unit. If you can, it may also be possible to add some filter material over the outside grill work. In my case, this made the biggest difference after simply weather stripping the perimeter of the case. I used window screen frame material cut to size and stretched it with a dark piece of fabric.
There's always working after dark, which a lot of folks resort too for other reasons. That would be a last resort, but keep it in mind.
When I win the lottery, I will replace the window unit with a mini-split ...
