I found some instructions that looked workable but I didn't have enough vinyl to do it without buying more. A check of TradeMe revealed that vinyl is surprisingly expensive. What to do?
Just then, Martin had to go out. He went outside then popped back in to pick up a piece of bicycle inner tube rubber, marvelling as he did so that they seemed to be an inexhaustible resource around here. Ah ha! I bet they'd make a nice wipeable cover for our stool :-)
A quick internet search revealed that others had had this idea before me. At first I thought I'd follow these instructions, but Martin thought a woven top would be nicer and I could always seal the edge with more inner tube, rather than metal. (Aren't the prices being charged for those commercially-available stools astonishing!!)
So, I set to and ripped off the old vinyl.
I pieced together some scraps of vinyl to protect the foam from any drips of water that might make it through the woven cover;
then wove strips of inner tube together in situ;
All up it only took me two 'up times' (a bit under two hours) and was immensely satisfying! Photos of the whole process are on flickr.