Police and fire crews were called to the A661 at Follifoot just after 7.30pm on Saturday (Feb 17) after a collision involving a Mazda and a Ford Focus.
Police said the driver of the Focus ran away from the scene of the crash.
The woman who was driving the Mazda suffered a minor head injury.
A North Yorkshire Police spokesman said air support was called in to help officers with the ground search, but the Focus driver was not found.
Enquiries are continuing to trace the driver.
Any witnesses are asked to call North Yorkshire Police on 101.