I've thought of this myself and these are my ideas which may or may not be helpful as I have only a little experience with vinyl:
Paint the base frame some color you like, either an accent to the vinyl or best match. This would be done so that if you don't get a perfect match between joints, you can hide it or make it a design feature. There are also parts that aren't practical to wrap, like drop outs or any exposed end of a tube. IMO, overlapping vinyl at the joints would look ugly (though a small overlap on the underside of tubes might be more durable and not well seen), so that's why I thought of this. Of course, you could overlap at joints and then carefully trim away the overlap to leave it perfectly flush, but I'm not experienced enough to know if/how much it will shrink back and where the overlapped vinyl had to step up from the tube, there probably would have been a little bit of stretching and the adhesion might not be optimal, so I considered it a potential peeling point.
Cut the vinyl for each tube with fish mouth ends as if they were the metal tubes themselves cut and flattened to be 2D. This will require some pretty good precision. If you don't have a computer program that could figure it out and print the patterns to size, it would probably be easiest trace each end with paper and transfer to the vinyl. With the program, you'd probably need to get the angles between tubes pretty close if you can't find the specs online.
Any braze ons would require some very careful planning and cutting around and, again, they could be pre-painted in a contrast/matching color and left or wrapped separately.
The rest would be skill in stretching and applying around various curves and keeping things aligned as you go. It would take some time and a lot of patience. If you go for it, I'd love to hear about how you ended up doing it.