This company "drilled" our borehole for water supply.
https://www.borehole-driller.co.uk/
Their rig is basically a steel frame tripod, an electric winch mounted on a truck, and a very very heavy piece of pipe. They drop the pipe repeatedly from the tripod and haul it back up. The soil and rocks get forced up inside. Our local rock is chalk, which is pretty soft.
A horse could definitely power the winch part, with the right mechanical advantage. I'd be sceptical about powering a rotary rock drill.