Fred Gipson's Old Yeller is full of conflicts, so the answer to this question could vary significantly depending on the reader and how specific one might want to get. The most evident conflict, however, is the conflict between man and nature.
Travis has been left in charge of his family while his father is away; he and his mother and younger brother “will be left in a wild frontier settlement to make out the best they [can].” Many things happen to make Travis's life difficult; among them is the arrival of a troublesome stray named Old Yeller.
Travis has countless struggles throughout the novel, and most of them involve his surroundings and the animals, both wild and tame, which he encounters. All of these constitute a conflict with nature. Time after time, Travis is confronted with life-and-death decisions, and all of them involve nature in some form. For that reason, the primary conflict in this novel is between man and nature in its various forms.