Yes, Travis did get his horse.

In chapter 1, Travis is waving goodbye to his father, who is getting ready to steer cattle to the Abilene cattle market. Travis is sad that his father is leaving but happy that his father will consider getting a horse for him. In exchange, Travis promises to fulfill his responsibilities (by taking care of his mother, little brother, and the chores) during his father's absence. Before he leaves, Travis's father maintains that what Travis really needs is his own dog.

For his part, Travis disagrees. The previous family dog, Bell, died from a vicious rattler's bite. Bell's death had completely devastated Travis, and he had then refused to entertain the idea of getting another dog.

Travis is utterly convinced that he will be far happier with a pet horse. He delights in daydreaming about riding his own horse across the plains. After Old Yeller joins the family, however, things change. Travis becomes attached to the dog, especially after Old Yeller saves his family from danger a few times.

In the last chapter of the book, Papa returns from his Kansas trip with a saddle horse for Travis. Accordingly, the "roan had all the pride and fire any grown man would want in his best horse, yet was as gentle as a pet." So, yes, Travis does get his own horse by the end of the story.