The Ford Mondeo was travelling in convoy with a Fiat Punto and a white Mercedes L200 on the eastbound side of the motorway when it left the entry slip road at junction 27 and crashed at about 3.30pm on Monday.
The men got out of the car and fled in the Mercedes, travelling east.
Officers are linking all three vehicles to an earlier incident at the Cottingley Springs site, in which several caravans were damaged.
Detective Inspector Phil Jackson, of Leeds CID, said: “We are investigating an incident of criminal damage in Cottingley Springs and suspect some of the people involved have crashed their car while trying to flee the scene.
“I am very keen to speak to anyone who saw the road traffic collision at junction 27 of the M62 and saw the people in the car abandon the Mondeo and then get into the L200.
“We have recovered the Ford Mondeo and enquiries are ongoing into the criminal damage incident.”
Anyone with information should contact CID on 101 or Crimestoppers on 0800 555 111.