Continuity mistake: When Marge sees Gaer shove Carl's body into the wood chipper, there's a huge blood puddle around the wood chipper. But moments later, as she shoots runaway Gaer, the wood chipper is visible behind her and the puddle completely disappears. There are only minor blood stains.
Continuity mistake: When Shep strangles Carl with the leather belt, as he is putting the belt around Carl's neck Carl's hands are caught inside the belt as the shot changes; but when the shot changes to a different angle his hands are seen outside the belt.
Continuity mistake: When the cop is shot in the head, no blood splatters onto Steve Buscemi's face, but there's blood on it in the very next shot.
Factual error: Per the opening credits, the movie is set in 1987, but when Carl leaves the airport parking garage, the billboard in front of him shows Northwest's "Bowling Shoe" livery which was not introduced until 1989.
Continuity mistake: When Steve Buscemi is shot in the cheek, he touches the wound, and in the first shot there's no blood on his hand, but in the very next shot there is.
Factual error: As a local Sheriff, Marge wouldn't be running an investigation into the shooting of a state trooper. She wouldn't have anything to do with it beyond responding to the initial scene. State investigators would investigate the death of a trooper, and since that event leads to the rest of it, the other events as well.
Character mistake: When Jerry and Scotty are talking in Scotty's bedroom, after his Mom's kidnapping, there is a poster behind Scotty's door. The poster says "Accordian King" but should have been spelled, "Accordion King."
Other mistake: When Jerry returns home to find that Jean has been kidnapped, he sees the shower curtain rod that's loaded with rings, and there are more rings on the floor. There's way too many rings for a single shower curtain that would use 12 or so at the very most.
Continuity mistake: When Marge goes back to Lundegaard's office for the second time, he is working on writing the vin numbers on a paper on a clip board, scratching them off and scratching his pencil on a "I love golf" notepad. When he leaves his office to complete a "Lot Count", Marge is looking around Lundegaard's office. The camera pans back to his desk and the paper that Lundegaard was writing the vehicles vins on the clipboard is no longer there and has been replaced with another piece of paper/application.
Answer: While there is nothing in the film that explains this, the most likely step taken by the authorities would have been to see whether Jerry and his wife had made out an will and who they would give legal custody of their son if anything happened to them. If there was no will then it's likely the closest family or relatives would be asked to take custody.
Lummie ★
He probably still has Lundegaard Grandparent (s) and a grandmother from his mother's side - any of which could and would become his legal guardian (s).
KeyZOid
He'll be a rich man. Grandfather said early in the movie that his daughter and her son would bever have to worry about money or words to that effect.