You would need to place a claim for the damage to your vehicle from hitting a deer under the comprehensive portion of your physical damage coverages.
No-Fault is a personal injury coverage and does not pay for auto body repair of your car or damage to any other party's motor vehicle or real or personal property. No-Fault is also primary to health insurance, which means it pays first in the event injury is due to an auto accident.
No-Fault, also called Personal Injury Protection (PIP), is designed to pay promptly regardless of who might have been at fault or whether there was any negligence, for economic losses (meaning medical/health expenses, lost earnings, and other reasonable and necessary expenses related to injuries sustained), up to $50,000 per person ("basic No-Fault coverage"), to the driver and all passengers injured in your car as well as any pedestrians injured by your car, because of its use or operation in New York State.
The purpose of No-Fault insurance is to restore individuals hurt in auto accidents to health and productivity as swiftly as possible. Because of New York's No-Fault law, lawsuits due to auto accidents can be brought only for economic losses that exceed No-Fault benefits and for non-economic damages (such as pain and suffering) only if a "serious injury" (as defined in the Insurance Law) is sustained.
As you can tell from this information from the New York State Insurance Department your no-fault insurance could only be used if you suffered injuries when your vehicle struck the deer.
If you need to place claims for the damages to your vehicle, under comprehensive coverage, and injuries, under no-fault PIP coverage, then call your agent or the claims division of your insurance company. The agent or claims department should be able to tell you the steps the take to place your claims.