The incident happened at about 4.10pm on Thursday at Ferry Road in Montrose.
The 43-year-old was working underneath a Citroen Relay van when the vehicle moved and trapped him.
He was taken to Ninewells Hospital in Dundee for treatment to serious injuries, with his condition described as critical but stable by medical staff.
Officers are now appealing for witnesses to the incident.
Sergeant Ewan Pearce said: “We would urge anyone who witnessed the incident or who has any other information to come forward.
“If you were driving in the area and have any dashcam footage or any private CCTV then we would ask you to review this and pass on anything of note to police.
“Any information can be passed to police by calling 101, quoting incident 2305 of April 22.”