The 76-year-old woman was walking along Littlemoor Lane in Balby, Doncaster at around 9.10am on Monday (March 29) when she is alleged to have been hit by the vehicle.
The woman was rushed to hospital with serious injuries to her brain and pelvis.
A 38-year-old man has been arrested on suspicion of theft of a motor vehicle, causing serious injury by dangerous driving and failing to stop at a collision scene. He has since been released under investigation.
A 56-year-old man has also been arrested on suspicion of perverting the court of justice and remains in police custody.
A South Yorkshire Police spokeswoman said: "Officers from our serious collision's unit are appealing for information, witnesses and dashcam footage following a hit and run in Doncaster.
"Yesterday at around 9:10am, officers attended reports of a road traffic collision on Littlemoor Lane in Balby involving a white Ford Transit tipper and a 76 year-old woman.
"It is believed that the Transit collided with the woman whilst attempting a reversing manoeuvre. It is alleged the driver and passenger got out of the vehicle and witnessed the woman on the floor before leaving the scene."
Anyone with information is urged to call police on 101 quoting incident number 209 of March 29.