Can I get a DUI riding a Horse in Ohio? Is a Horse considered a Vehicle?

No, you cannot get a DUI on a horse in Ohio...

...unless the horse is pulling a vehicle, such as an Amish buggy.  Ohio revised code states that "No person shall operate any vehicle, streetcar, or trackless trolley within this state" while under the influence.  If you are controlling the horse pulling the buggy, you can get an OVI.   As a passenger in the Amish buggy, probably not.

If you are merely riding a horse without a wheeled carriage, it is not considered a "vehicle."   "Vehicles" means everything on wheels or runners, including motorized bicycles, but interestingly does not mean electric personal assistive mobility devices, vehicles that are operated exclusively on rails or tracks or from overhead electric trolley wires, and vehicles that belong to any police department, municipal fire department, or volunteer fire department, or that are used by such a department in the discharge of its functions.   A vehicle does not include a Horse.  

It is still not a good idea to ride a horse while intoxicated due to the risk of death or serious bodily harm.  And, you could be charged with other offenses if you are endangering other people, particularly on a roadway or in a public place.