A father and his two young sons were airlifted to the Bath Royal United Hospital with injuries that are not thought to be life threatening, the BBC reported ambulance staff as saying.
Medical treatment was given to a fourth person at the scene, while the driver of the car suffered from shock but was not injured.
"The crowd was probably 20 to 30 metres away from the track (but) the car carried on going its way into a young family, two young lads, eight or nine years old, and their dad," an anonymous fan was quoted as telling the Wells Journal.
"I saw the young lads fly up into the air. The dad flew over the top of the car and then it came to a stop and the driver was in a bit of a state, psychologically, kicking the car, he was very upset about what had happened."
The local newspaper later quoted ambulance staff as saying the injured adult male had suffered a head injury but that it was not of grave concern, and that his sons had suffered leg injuries.
The Regency Stages Rally at the Royal Bath and West Showground will continue after organisers made changes to the track, Avon and Somerset Police said.
The incident happened at around 11:50am UK time. A joint police and Health and Safety Executive investigation is taking place.