Was the original not some sort of rubber? It is usually a form of rubber with a fabric core, but I doubt its feasible to reproduce that sort of fabric yourself without access to industrial equipment.
But I think you can use the cotton fabric - builders of Greenland kayaks do this all the time, and so far as I know they use some sort of varnish to "dope" the cotton and make it waterproof enough for immersion. My worry would be whether a fabric treated this way would stand up to repeated folding and unfolding - I just don't know.
You could try the QajaqUSA forums
for more information on this approach.
Otherwise, you could go with PVC, and for that the best source of information on that would be Tom Yost's site