No, it is not legal to possess a deer that was hit by a car in New York State. According to the New York State Department of Environmental Conservation (DEC), it is illegal to take, possess, or transport any wild animal that has been killed by a motor vehicle, unless such animal is taken by the owner or operator of the motor vehicle that killed the animal, or by a person acting on behalf of the owner or operator. Additionally, it is illegal to possess any part of a deer that has been killed by a motor vehicle, including the meat, hide, antlers, or any other part.
If you find a deer that has been hit by a car in New York State, you should report it to the DEC by calling their Wildlife Hotline at 1-800-847-7636. The DEC will then send a conservation officer to investigate the incident and determine if the animal can be salvaged for consumption. If the animal is not salvageable, the DEC will dispose of it properly.