The plane came down near the village of Churchinford yesterday on its way from from Surrey to Dunkeswell aerodrome, an airfield in east Devon, police said.
The family members, from Surrey, were a 56-year-old man, 55-year-old woman, 23-year-old woman and a 20-year-old man.
Superindent Richard Corrigan said "our thoughts are with [the affected family] at this extremely difficult time".
Crews from the Devon and Somerset fire and rescue service and the South Western ambulance service joined police at the scene.
Formal identification has not yet taken place and a joint investigation with the Air Accidents Investigation Branch is under way.
Corrigan added: "The focus of the activity will be to investigative inquiries and then recover the aircraft.
The fire service deployed 27 firefighters in six vehicles, and the ambulance service dispatched two vehicles, an officer and the Exeter-based hazardous area response team.
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