Afaik most drivers in the Prussian service were civilians. The same in the austrian army. It is possible that drivers received a military coat to be clothed in more uniform way
I have some four pictures of Austrian drivers of 1762.
First team is drawing a gun, second time a ponton, third time a field bakery and the fourth time a wagon with provisions.
Except in the case of the bakery, there are four horses and the driver is on the rear left horse, the bakery is pulled by three horses (front one, rear two and the driver is on the left rear horse)
The drivers of the first two (gun and ponton) is wearing grey coats with yellow collar and cuffs but grey turnbacks, straw trousers and black boots. The black tricorn has yellow lacing. The pontoon rider has a yellow band (brassard?) around the arm with black austrian eagle
The other two drivers have blue uniform and yellow cuffs, turnbacks and collar, black tricorn with yellow lace, strawcoloured toursers and black boots. Both have yellow brassards with black eagle