The first reliable reference I can find for the use of Bus as an ambulance in NYC dates back to 1903…long before GM, or Grumman. The mode of ambulance propulsion was four legs.
If you read the autobiography of Dr. Emily Dunning Barringer- “Bowery to Bellevue, The story of New York’s First Woman Ambulance Surgeon”, you will find it was a common reference to the horse drawn carriage ambulance of the day. How it came into existence is still a mystery, but it is thjought to come from the word omnibus.
Calling an ambulance a bus is a tradition that has survived more than a century. Its use a celebration of tradition. It honor the ambulance men that came before.
(Random House Historical Dictionary of American Slang, Vol. I A-G)
bus n. (...)
2. an automobile or other motor vehicle.
1914 T.A. Dorgan, in Zwilling TAD Lexicon 22: A friend of mine just bought a new car. A flivver...Thats like mine - some bus.
1916 W.J. Robinson At Front 121: The old 'bus made the most of what she had.
1917 Imbrie War Ambulance 115: A car was a "buss."
1917 in Dos Passos 14th Chronicle 92: Our ambulance however is simply peppered with holes - how the old bus holds together is more than I can make out.
More information on Dr Dunning:
Dr. Emily Dunning Barringer
Dr. Dunning Riding the Bus-
from the movie "The Girl in White" with June Allyson, Jessie White, and James Arness
If you would like more information on the history of EMS in New York:
FDNY/New York City EMS History