How are children's board books made commercially.  What equipment is
used and what is the process?

Well you need a press that can handle cardboard.
Might need to coat the cardboard.
Binding might be tricky, since book will be chewed on and tossed accross the room.
Special knives to cut the shape of a duck, etc.
Sounds like a special bindery with special equipment, long since written off.
They made board books back in your great Grandma 's time.