Although there was not a veterinary school in Kirksville, the University of Missouri-Columbia began offering courses in veterinary medicine in 1872. In the mid- to-late 1800s, there were other schools in the area teaching veterinary medicine. The St Louis Veterinary College opened and closed in 1875. The Kansas City (MO) Veterinary College opened in 1891 and closed in 1918. The Western Veterinary College (Kansas City MO) opened in 1897 and closed in 1908.
It was also customary at the time to apprentice, either to a person who was already treating animals, or to one of the graduates of one of the European veterinary schools who had emigrated to the US. There was a Lyman Brown practicing in Hamilton at that time, but I believe that's at least 100 miles from Kirksville. There doesn't seem to be anyone who identified himself as a veterinarian, a veterinary surgeon, or a farrier in Kirksville in the 1870 census, but you might want to look at the 1880 census, to see if there was someone with whom your ancestor may have apprenticed.