It has been a long time since veterinary college teams took the field, but that time existed in football's past, just like the business college and medical, dental, and law school teams that existed before 1910.
An RPPC that turned out to show the Chicago Veterinary College football team led me down this rabbit hole, so I wanted to share my pet project findings with you.
Most professional schools were not affiliated with traditional colleges and universities at the turn of the last century. Hospitals hosted medical schools that functioned like apprenticeships, and there were many independent medical, dental, and other professional schools, including veterinary schools.
As I snaked my way through the history of veterinary schools in the U.S., I learned that pre-WWI veterinary schools focused on caring for large animals. Besides meat and milk-producing cattle and sheep, the country had many more horses hanging around. Henry Ford and others changed that, so there was less need for veterinarians until the veterinary schools refocused on pet care. Fewer veterinary schools also meant fewer veterinary school football teams.
The Chicago Veterinary College, also known as CVC, existed from 1883 to 1920, topping out at 400+ students. Back then, Chicago's numerous professional schools faced one another on the gridiron. While not part of an official conference, the colleges teaching various forms of medicine competed for the medical school football championship. Head-to-head battles and the transitive property allowed one school or another to lay claim to being the top dog at season's end.

A close review of the RPPC below shows that CVCs had two primary rivals: the Northwestern Medical and the Physicians and Surgeons teams. A common practice then was to take a team photo at season's end, often with the team captain holding a game ball with the year and key scores painted on. The RPPC below does not show the year but provides three game scores.
Expanding the image reveals that the CVC team beat NVM (Northwestern Medical) 10-0 before taking down Physicians and Surgeons twice by 5-0 scores, making them 3-0 against their rivals, so they likely claimed the Chicago medical school championship that year.
Rivals, of course, are teams that have some much in common that they come to hate one other, so it is important to choose your rivals carefully. While Northwestern Medical was a primary rival, they were a disorganized team.
For example, consider the CVS-Northwestern Medical game of 1905. Normally, the teams played before small crowds, but they scheduled the 1905 game as the curtain-raiser for the Chicago-Northwestern rivalry game. They scheduled the vets and medics game for a 12:30 start -two hours before the varsity contest- but the Northwestern Medical team arrived late, so they only played two ten-minute halves. The same day and down the road in Champaign, Illinois faced the Chicago Physicians and Surgeons team, who they dissected for a 30-0 victory.
The following week, the vets were the curtain-raisers again in a medical school doubleheader at the American League park. The vets faced the always-tough Bennett Medical College team, after which Northwestern Medical faced the Chicago Dental School.

The third article in the image above shows that even the Chicago Academy of Fine Arts fielded a football team, though the artists were starved for opponents.
While Northwestern Medical's late arrival for the 1905 game showed you can never trust your rivals, things got worse in 1906 when they did not show up for the CVC game. Apparently, someone locked up Northwestern Medical's uniforms so they were unable to play the game.
Besides playing professional schools, CVC faced a mishmash of opponents. Some years, they played and lost to local high schools while meeting and beating small colleges, military teams, and anyone else who could get eleven guys on the field. They even scrimmaged the All-Western All-Stars as the All-Stars prepared to play the professional Massillon Tigers. The All-Star team, led by former Michigan star Willie Heston, likely beat the vets like rented mules since the next day they lost a close game to the Ohio League champions, but only after Heston broke his leg.
I could go on about the CVC team, but rather than beat a dead horse, we'll end our story here. However, I'll leave you with one thought or question. Since many teams use animals as their mascot, what is an appropriate mascot for a veterinary college football team? A comparable team in NYC would obviously be the New York Vets, but your suggestions for Chicago’s team are welcome.
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Top dog! Like rented mules! Beat a dead horse! New York Vets! This is your most animated item ever!!
Turning things back to my own team (as I'm wont to do), Willie Heston was NC State's first major brush with football fame. Heston coached the Farmers during the 1906 season you mentioned, but left Raleigh about 10 days before their final Thanksgiving Day game vs Virginia Tech, taking a $100 (about $3k now) pay-cut to do so. Heston was originally supposed to participate in a Michigan alumni vs Chicago football game, but it was canceled last second. The regrouped and participated in the All-Western All-Stars - Masillon Tigers game instead. I never knew they had a tune-up game before.
Heston had remained notably skeptical the new rule changes of 1906. His conservative play style (eschewing new open formations and instead preferring old-style kicking or mass plays) led the Farmers to their most ties in school history, finishing the season 3-1-4 despite considerable perceived talent returning to campus. He doubted the changes improved player safety, remarking to Michigan training staff pre-season "You'll have to train a squad on how to carry stretchers." His doubts were likely reinforced by the injury he sustained in the Masillon-All Stars game, after which he swore to never play football again; as best I can tell, he held true to that claim.