If the holes can be perpendicular to a plane that bisects the sphere of your golf ball and parallel to one another, and you're just talking about the three holes that form an equilateral triangle on the right side of your picture, that's not too hard. Use a brad-point drill bit if it works better than a router bit; the only reason you'd need a router bit is if you were cutting sideways, but that doesn't seem to be the case. Use your router to build yourself a fixture which will hold the maximum number of balls in a regular array, and set up your G-code program to address them sequentially.
If the holes need to all point towards the center of the spheres, that's much harder; you'd either need a 5-axis machine to tilt the spindle and address each hole from a different direction, or a way to reposition each ball between drilling each series of holes. That could be done with a fixture furnished with pegs that fit in the first set of holes and registered them into position for the second set, etc.