60 or higher pound cardstock may work even w/o the supports, just experiment with smaller pieces before going for a large job. If you want to be sure, add them. Like BFD said, even regular printer paper could work.
I've noticed a discrepancies between the actual stiffness/thickness of cardstock between the different suppliers I've used (not all for Pepakura, some for class and extra-curricular projects). Some of the stuff labeled as 110 behaves more like another company's 98 or lower, and this generic stuff I got from Sam's is like 110+.
Another thing that causes difficulty during printing is paper feed angles. Top-feeding trays have worked better for me than bottom tray fed (where the paper has to flex ~180 degrees) supplies, excepting the one industrial strength printer I used (and had to un-jam 3 times) at work.