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You will need to obtain a duplicate NY title through the seller. If you cannot find the seller, then it becomes more complicated because you would need to go to court in NY to file a petition to quiet vehicle title and provide your proof of purchase and would have to seek to serve the prior owner a copy of the suit and then go to court to have the court to order NY DMV to issue a title in your name.
I am afraid the seller has to apply for the duplicate title or you have to get a court order in NY to issue title in your name upon proof of purchase and you can seek the costs and damages from the private seller for having to do so, since they have a legal duty to provide you the title. So when you sue to quiet title, you would seek damages from the seller for failure to provide the title in the first place.
A local attorney in NY can perform this for about $1000 or so, but again, you can recover those costs from the seller for not providing you proper title to the vehicle. I would still try to contact the seller first to get them to apply for duplicate title and then have them sign it over to you as should have been done in the original sale. If not, get a consumer protection lawyer in NYC from the same sites used by other attorneys, http://www.hg.org or http://www.lexmundi.com