The incident was reported in front of 108 Locust Avenue in the New Dorp section around 12:30 p.m.
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Authorities said the victim, who lives in the vicinity of the crash, was getting out of her vehicle from the driver's side when a black Chevrolet sedan struck her.
She was taken to Staten Island University North hospital with a serious leg injury, and she has since been upgraded from critical to stable condition.
The suspect, a 29-year-old woman, apparently jumped out of the car and ran away.
Detectives say they have a good idea of her identity and she was traveling at a significant speed at the time of the crash.
The vehicle and a passenger inside remained on the scene.
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No arrests have been made, and the investigation is ongoing.
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