You don't really want to replace the material with a knitted bag as it will not successfully contain your knitting equipment without a woven fabric lining.
I picked one up at a tag sale and have been meaning to replace the bag, so I just took a good hard look at it. The bag part was definitely assembled and put on the frame before the frame was glued together. The bag is also much more complicated that I previously thought, having an extra sleeve which wraps around the bottom horizontal rods of the frame so when you open the bag it takes on the shape of fully open brown paper grocery bag.
There's no way to attach a new bag with a normal household sewing machine, so I think unless you feel like doing a lot of sturdy hand sewing, the best bet would be to make a new bag using the old one as a pattern, but sewing on velcro strips to attach it to the frame. As long as the velcro is well covered, your yarn shouldn't snag on it. Or, you could use sturdy snaps. Then, at least, you could take the bag off for washing. The frame on mine is lovely, but the fabric part is ugly and torn, and honestly, I'm a clutz and need one I can wash.