Hey Jake,
Finding commercial printers who can handle the production of pop-up material is a challenge. I don’t even know of any off-hand in my neck of the woods, so I can’t really help with a name or contact.
As a general guideline, it’s best to go talk to a few printers and see what they are capable off and how much they are willing to do. Theoretically, any printer who can manage detailed ‘die-cuts’ and has enough man-power to assemble things with some skilled humans at hand should be able to pull it off. But best to judge these capabilities yourself with either samples or confidence in the people in charge.
Samir