How to support heavy weights when constructing chairs with cardboard?
You don't specify what kind of processing is permitted on the cardboard, if any, but if you have relatively free rein I'd recommend investigating papier mache - it has been used historically for furniture making so could represent a very viable option for making chairs out of cardboard.

Options range from a very basic and functional version made from essentially cardboard tubes glued together, to very ornate items such as seen here:



From papiermache.co.uk

At the most complex end of the scale, paper and cardboard can be glued in layers to make a material which is almost as strong as wood, and can be cut, carved and sanded in much the same way.  This is very time consuming and can require specialist equipment so it's not particularly practical for hobby applications, but it goes to show the kind of strength that can be obtained.