The University of Indiana library's online exhibit has a letter by Henry Dearborn "to John Langdon, D. Commissary, General of Purchases. Gives explicit instructions for the purchase of horses for the army."
Commentary from Kentucky about the effect of war on horses and horse racing:
Q: What hurdles — pardon the pun — did horse racing overcome to become dominant in the Southeast United States?
A: War, corruption, religion and geography. The War of 1812 decimated equine herds, and social reformers essentially shut down racing in the North and East. But Lexington kept its track open.
There is a Society of the Military Horse message board here with a few War of 1812 references. Here's an archaeology picture of a double horse burial at Fort Meigs.