Yes, In accordance with NY State law anyone using this method has to apply to DMV and sign an agreement identifying their permissible use under the DPPA. Searches cost $7 each. Searches are usually done by insurance companies, who have to rate and monitor drivers or investigate claims. Searches are also done by employers who are required by monitor their employers driving records.
I just remembered that I need to ask about buying a vehicle from another person in a private sale.  If I recently bought a vehicle from a previous owner, is there anything I need to do in regards to this?
You need to get the car's taxes calculated. For that, you and the seller need both to fill a form DTF-802,  so the DMV can do the taxes maths.
Do you know about the DPPA. Is there a problem if the vehicle was registered for me in another state while I was living there?