This applies much more to modern armies, which have notoriously long pack tails supporting relatively few fighters. Napoleon had more fighting men available to him than the US Army did in World War 2, for instance.

In October 1862, the Army of the Potomac had 3,219 wagons and 315 ambulances in service. The standard allotment, as established by order of Halleck on August 21, 1863, was 20 wagons per 1,000 men. If we assume one teamster per wagon or ambulance, that's anywhere from 2%-4% of the army's fighting strength. Certainly blacksmiths, armorers, supply and medical personnel would add to that somewhat, but I cannot envision them ever equaling more than 10% of the army's fighting strength. Supplying a Civil War army was always extremely difficult when away from a railhead. For every wagon with its requisite team of mules horses you brought, you increased the amount of forage the army had to carry; what might be described as an example of the law of diminishing returns. Thus, they attempted to keep supernumeraries to a minimum for that reason.

Another group consisted of the sutlers, who were not officially part of the army, but existed to sell to (read: defraud) the men of the army. On April 21, 1864, Grant expelled 2,800 of them from his area of operations, thus giving an idea of their numbers.