Interesting question. I was thinking about how to cut a print into a puzzle without getting too carried away purchasing equipment. I spent a few minutes searching for puzzle dies, but the only one's I could find in the 500-1000 piece variety come from China: http://www.puzzlemakingmachine.com/standard.html
But I think if you could find the right die, you could probably generate enough pressure to cut through cardboard with either a laminator or an etching press (or obviously a die cutter, but those get expensive/heavy/big in the size you are looking for). Not sure if you have a laminator (or know someone who does), but my suggestion would be to buy a cheap die, for example: http://www.dickblick.com/1/1/23992--allstar-die-cutter-machine-puzzle-1-die.html
and see if it cuts through cardboard (or whatever you'd like to use as the puzzle material) when you pass it though the laminator. If that works, it might be worth navigating the treacherous waters of international trade.