It happened on May 18 at 52nd and Catharine streets around 9:39 p.m.
The victim's niece, Sandra Green, identified the woman as 63-year-old Antionette Smith.
Green said Smith is on life support at Penn Presbyterian Medical Center.
"Turn yourself in because that could have been one of your family members. Would you let your family member stay out in the street like a piece of trash?" said Green.
Surveillance video shows a blue Can-Am Maverick traveling north on 52nd Street from Baltimore Avenue when it entered into the intersection and struck Smith.
The vehicle continued north on 52nd Street and turned eastbound on Cedar Street.
Video showed the reckless driver passing vehicles on the wrong side of the road.
"This is a Can-Am vehicle that's made for off-road use," said Captain Mark Overwise with Philadelphia Police Accident Investigation Division.
Green said many in the neighborhood affectionately called Smith, "Buttercup," and some area police officers said they knew her well.
"Whoever hit her, left her there for dead," said Green.
The family is pleading for the driver to come forward.
Police said the vehicle is so unique that someone in the area must recognize it.
Anyone with any information is asked to call police at 215-686-TIPS.