The suspect used an accelerant to ignite the Audi A5 outside Deli 29 in Keighley Road, Bradford.
Police believe that the suspect was caught in the fire as the car went up in flames.
Following checks at hospitals officers are now appealing to people who were in the area on Wednesday, January 8 to help with this investigation.
Investigating officer PC Dan Holdsworth, said: “We believe this person would have suffered burn injuries which may need further treatment, so it is important that we locate them firstly to make sure they have received appropriate medical care.
“We are appealing to anyone who was in the area at around that time and particularly to motorists, as they may have dash cam footage which can assist our enquiries.”
Anyone with information is asked to call police on 101, or use the contact options on the West Yorkshire Police website.
The crime reference is 13200013446.
Information can also be reported anonymously to the independent charity Crimestoppers on 0800 555111 or online.