The body of Joe Boever, 55, was found the next morning, authorities said.
Victor Nemec, Boever's cousin, said in an interview with the Argus Leader that the attorney general "hit my cousin as he was walking down the side of the road and killed him".
It's not clear why Boever apparently chose to return to the truck on foot on Saturday night instead.
Nemec told ABC affiliate Dakota News Now he hasn't been contacted by the state and he believes it is being tight-lipped about the investigation.
"It seems like there has been a lot of foot-dragging and lack of investigating going on", Nemec told the station.
Mr Ravnsborg released a statement on Sunday night addressing the incident.
In Ravnsborg's own words, he's "shocked and filled with sorrow following the events of last night. We have no reason to doubt that he did not continue that trend", he told ABC News.
"We are now working the investigation as referenced by Governor Noem in Hyde County", Price said Sunday.
He added that was co-operating with the investigation, while offering his "deepest sympathy and condolences to the family of Mr Boever". A spokesperson for the South Dakota Department of Public Safety said details about the vehicle accident, like whether alcohol was a factor and if any charges are pending, are expected to be shared on Monday.
Governor Kristi Noem said the state's Highway Patrol will conduct the investigation into the crash.
Ravnsborg's office says the Republican Attorney General was driving back from a Lincoln Day Dinner at Rooster's Bar & Grill in Redfield and he had not been drinking. He pleaded guilty to speeding six times between 2014 and 2018 and paid fines between $19 and $79, according to state records. He also received tickets for a seat belt violation and for driving a vehicle without a proper exhaust and muffler system. They both returned to the accident scene, where Volek said he would handle the investigation and asked Ravnborg to return to Pierre, according to Ravnsborg's statement.