The incident happened at the family business, which appears on Yesterday show Bangers and Cash, on July 26 outside their showroom and museum in the village of Thornton-le-Dale, near Pickering.
The fire was started deliberately after an accelerant was thrown.
North Yorkshire Police said: "The incident happened at Mathewsons in Thornton-le-Dale at around 1.40am on Tuesday 26 July.
"One vehicle was damaged but fortunately the fire did not spread to other cars, the garage itself and nearby buildings where people were asleep inside.
"Officers are already exploring numerous lines of enquiry and believe the offender may have sustained burn injuries to their arms, but this is not confirmed."
PC Alan Robson, who is investigating the incident, said: “Thornton-le-Dale is a quiet, peaceful Ryedale village and the Mathewson family are very keen to establish who was responsible.
“We already have several strong lines of enquiry and we are now requesting the public’s assistance to track down those responsible.”
Anyone with information that could assist the investigation should email [email protected]
You can also call North Yorkshire Police on 101, select option 2 and ask for PC Alan Robson.
Mathewsons was established in 1970 in Bedfordshire before a move north in 1988 and the purchase of garages in Scarborough, Malton and Pickering. The Thornton-le-Dale site opened in 2011.