Yes. You can apply for a credit when you apply for registration in New York. However, you can only be eligible for this credit if New York and the state where you paid sales tax have a reciprocal sales tax agreement.
III didn't buy from a dealer. I bought from a private individual. What would be considered an acceptable proof of ownership in my case?
Since the atv was transferred by a private individual, an acceptable proof of ownership would be a New York State or out of state transferable registration or title certificate signed by the previous owner to transfer ownership to you.