Trivia: The movie wasn't based on a true story. The Coen brothers just threw that in. That said, the special edition DVD of the film contains a statement that the film was inspired by a real life incident in which Richard Crafts killed his wife Helle and disposed of her body by feeding it through a woodchipper.
Trivia: When William Macy was trying out for the part of Jerry Lundegaard, he said to the Cohens he'd shoot their dogs if they didn't give him the role.
Trivia: Note in the end credits that there is the symbol for "the Artist Formally Known as Prince" (a Minnesota native) placed sideways.
Trivia: The storyboard artist for the movie has a small role. He is the driver of the car that slowly passes by when Buscemi is dragging the state trooper's body off the road.
Trivia: Peter Stormare only has 18 lines in the entire film.
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.