I used to live on Staten Island and the DMV was always easy to deal with. I registered several 'paperless' cars with their help. I'd try making a personal visit, explain your situation, and see what they can do for you.
Is there an ID number on the car that you can find? Does it have registration stickers on the windshield? New York registration stickers have had the VIN and plate number printed on them since about 1980, which would help you considerably.