Old Man Fat keeps threatening Chee with the government man in town. "Who" is the government man, and why would he have any power over Chee?

From the story, the "government man" quite possibly refers to the state authorities (who are usually backed by the courts in cases involving domestic conflicts). Old Man Fat is saying that government officials have the jurisdiction to decide who Chee's daughter lives with, and he's warning Chee against taking his daughter by force. Old Man Fat's implication is that Chee could be arrested or prosecuted if he tries to get his own way.
Chee knows that the authorities in his own tribe would side with Old Man Fat, and his father-in-law knows this as well. After all, the tradition has always been that a daughter belongs to her mother's people. Because of this, Chee feels that he has little hope of getting his daughter back no matter which authorities decide his case.

He would have to give his daughter up if the case were brought before the Headman of Little Canyon, and certainly he would have no better chance before a strange white man in town.

In the end, Chee comes up with a novel way to change his father-in-law's mind. He plants two fields: a large one, filled with corn, pumpkins, and squash, and a smaller one, filled with onions, carrots, and chili peppers. At the end of the summer, he reaps a bountiful harvest and ties bulging packs of food across two pack ponies to take to his in-laws. When he sees Old Man Fat again, he notices that the older man has lost some of his previous cockiness.
Apparently, the trading post has closed, and Chee's in-laws are now struggling financially. The story ends with Chee unloading a winter's worth of food in his in-laws' home; with the food, he manages to reclaim his daughter from her grandparents.