Another great cardboard resource is the tubes that come in the center of rolls of plotter paper. Check out architecture/engineering/art studios on college campuses, behind art supply stores, or at local printing companies. On another scale, cardboard tubes used for concrete formwork are available at hardware stores. Shigeru Ban, a Japanese architect, has built a number of bridges, houses, and other structures out of paper tubes. They are incredibly strong, and take screws well.
The first chair is made from tubes for the seat and backrest, and more tubes are buried in those side pieces to make the support structure. The corrugated cardboard is screwed into the tubes with drywall screws to make essentially a giant gusset that holds them all in place.
This second chair is made from shipping tubes. I wrote a whole instructable on this one, found here: FedEx Stool