I spent a lot of time looking for the answer to this question, and I must confess that O could not find it, either through reading the text or checking other sources. Here is what we do know about Carl Linstrum in O Pioneers! by Willa Cather.
First of all, we know that Carl and his family moved from St. Louis, Missouri, to homestead a farm in the Nebraska plains. In the ten years or so that they live here, the farm is not financially viable. At the end of ten years, Carl visits the girl he loves, Alexandra Bergson (of Swedish descent), and sadly announces that he and his family will be moving back to St. Louis where his father will once again work in a cigar factory.
Carl sadly tells Alexandra:
"I am going to learn engraving with a German engraver there, and then try to get work in Chicago."
Earlier in the novel, we learn that Carl has drawn some things with specific connections to other countries, including
"hunting pictures in Germany, and Robinson Crusoe and funny pictures about cannibals. I'm going to paint some slides for it on glass, out of the Hans Anderson book."
So, we have two references to Germany, one to the Danish author Hans Christian Andersen, and one to an English novelist. None of these give us a clear indicator of Linstrum's origins.
Sixteen years after he leaves, Linstrum comes back through Nebraska on his way to Alaska for the Gold Rush, which also does not help determine his heritage. One variation of the name (Lindstrom) is Swedish, but Linstrum is not connected to any specific heritage, as noted in the ancientfaces link attached below. The eNotes characters site attached below refers to him as a "transplanted European," but no specific ethnicity is mentioned.
The general presumption of the characters in this novel is that all of the farmers were Europeans who came to America for a better life. The Bergsons and Crazy Ivar are Swedish, the Shabatas are Bohemian, Amedee Chevalier is French, and there is a Norwegian cemetery in the story. Clearly there is a pattern of European immigration; however, I could find no substantive reference to the Linstrums' nationality.