An ambulance was originally not a vehicle at all. It was a hospital that walked.
The word comes from the French phrase (hôpital) ambulant, a "walking hospital," which Etymonline dates to the 1600s. That in turn comes from the Latin ambulare, "to walk, go about." Etymonline dates the English word to 1798, when it meant a mobile or field hospital. It became a vehicle for carrying wounded soldiers off the battlefield in 1854, during the Crimean War.
The vehicle idea owes a lot to French military medicine in the 1790s. Dominique Jean Larrey, who later served as a leading surgeon of Napoleon's armies, was troubled at the battle of Spires. Wikipedia's history of the ambulance says the ambulances stayed far from the fighting, and the wounded were collected only after combat ended.
His fix was a horse-drawn wagon crewed by trained drivers, corpsmen and litter-bearers. Wikipedia says Larrey adapted the axle design of horse-drawn artillery, which handled rough ground well. He called his wagons ambulances volantes, or "flying ambulances."
The history page says the Army of the Rhine first used them in 1793. Larrey later adapted the idea for other campaigns, including a camel-borne litter for the Egypt expedition of 1798 to 1801.
Notice what the word kept. The French phrase had two parts, "hospital" and "walking." English dropped the hospital and kept the movement. By the time motor vehicles arrived, the name had already moved from the place where care happened to the thing that carried the patient to it.
The motor ambulance came late. Chicago's Michael Reese Hospital received what the history describes as the first automobile ambulance in February 1899. It calls James Cunningham, Son & Company's Model 774 of 1909 the first mass-produced one.
The word also gave English an insult. A lawyer who hunts for accident victims is an "ambulance chaser," a term Etymonline records from 1897.