It happened on Sunday at around 12.20pm on Manor Drive, just off Otley Road near the Cooperative store, as the 82-year-old woman crossed the road.
Officers said the vehicle, described as a small white hatchback, had left the scene of the incident before police arrived.
The woman was taken to Leeds General Infirmary with face injuries and has since been discharged.
Police are appealing to the people who were in the car to contact them, along with anyone else who witnessed the collision, or anyone who has any information that would help identify the vehicle and its occupants.
Anyone with any information that could assist the investigation is asked to contact North Yorkshire Police on 101.
Quote reference 12180055098 when passing on information.