According to the New York State Department of Motor Vehicles you are not required to have a driver license or learner permit to apply for a vehicle registration or title certificate in state of New York. These requirements also apply to the registration of a boat, a snowmobile, or an ATV.
So without a valid license you can buy, register and title a car in your name however getting insurance will be more difficult. Your driver's license record is one of the main rating factors typically for any car insurance carrier so without a valid license they do not have a motor vehicle record to rate you on. Or if your license is not valid due to it being suspended or revoked than an insurer will have difficulties wanting to insure you since you are a risk to them. Obviously if your license is not valid you are not supposed to be on the roadway with the car and so you driving without a valid license pose liability risks to your insurance provider that they do not want.
If you do not have a valid license due to a medical condition, disability or even due to a suspension or revocation you may still be able to get insurance for a vehicle if you are excluded as a driver from the policy and instead have the person (or people) that are going to drive you around in your vehicle listed on your auto insurance policy as the drivers. The car insurance company could then rate your policy using your driving record.
Not all insurers are willing to exclude the owner from their own policy so you may need to shop around for the right coverage. There are insurance companies, such as Esurance, that allow another person to insure your car, even without insurable interest in it so that could be another option where the driver is the person insuring the car instead of you the owner without a valid license.
Getting auto insurance without a valid driver's license may be possible but is a lot more difficult to do because of insurance company guidelines, not any NYS State laws. New York State just wants you to carry at least the state's minimum car insurance requirements on your vehicle if it is registered in NYS.