The driver of the orange-coloured car stopped at the scene of the collision in Albermarle Road and spoke briefly with the injured teenager but then drove away without leaving their details.
The boy is receiving treatment for his injury and is likely to require surgery to repair the fracture, police said.
Police are appealing for witnesses to the collision which happened at 1.50pm on Thursday, or anyone who recalls seeing the vehicle before the collision, to contact them.
A spokesperson for North Yorkshire Police said: “In particular, we are appealing for the driver of the vehicle in question to contact North Yorkshire Police as soon as possible.”
Anyone with information is asked to contact the force on 101, select option 2, and ask for Alexandar Saggers, quoting reference number 12180040048.