Here is a link to DIY CNC Milling machines
What you need is simpler as you only need to index your platter and move the drill up and down, but the general principles in the milling links should be helpful.
I think your main problem will be holding the corks firmly and verticaly.
Also if the corks are in firm enough for drilling it might be a pain to get them back out, though I suppose the hole might make that easier.
You might want to make the holes in the platter stepped.
The corks would be pushed into the platter till they hit the step.
After drilling you could press the platter onto a circle of pegs that would go up through the holes and eject the corks.