Yassar Yaqub, 28, was fatally wounded in Huddersfield on a slip road at Junction 24 on Monday evening.
What appeared to be a non-police issue firearm was found in the vehicle Mr Yaqub was travelling in, the IPCC said.
Police are believed to have searched his home in the town. Armed officers could be seen outside.
West Yorkshire Police said earlier that Mr Yaqub was shot during a planned operation.
The weapon found at the scene is being "further examined", the IPCC said.
Its investigators are continuing to seek any relevant CCTV footage and initial accounts have been obtained from the police officers involved.
A West Yorkshire Police spokesman said five people had been arrested as part of the operation, including three in the M62 incident.
Several of those arrested required hospital treatment for injuries "not related to firearm discharge", the police spokesman said.
No police officers were injured.
A spokesman for Mr Yaqub's family said they were "in shock, and are distraught". He asked for their privacy to be respected.
Mr Yaqub, of Rudding Street, Crosland Moor, Huddersfield, was acquitted of attempted murder seven years ago following a shooting in Birkby in the town.
He was found not guilty after a judge threw out the prosecution case against him, the Huddersfield Examiner reported.
IPCC Commissioner Derrick Campbell: "My thoughts are with Mr Yaqub's family and all those affected at this difficult time.
"Though in its early stages, this investigation is making good progress but will be complex." He asked for patience while the IPCC investigation continues.
The BBC's home affairs correspondent Danny Shaw said the M62 incident was the fifth fatal police shooting in England and Wales in the past nine months and the first involving West Yorkshire Police since December 2010.
November 2016: Lewis Skelton, 31, was Tasered and shot dead by Humberside Police in Hull, amid reports he was "running around with an axe".
November 2016: Josh Pitt, 24, died after being shot by a firearms officer in Luton as Bedfordshire Police responded to reports that a woman had been assaulted.
May 2016: William Smith, 36, was shot dead in Goudhurst, Kent, while on bail in connection with the death of 73-year-old Roy Blackman.
March 2016:  James Wilson, 24, died in hospital after being shot by Northumbria Police in South Shields after police responded to reports a man was holding a handgun.
A suspected firearm was found in a car carrying a man who was shot dead in a police operation off the M62, the watchdog the IPCC has said.