I think they do, I just checked Harriet Carter and Walter Drake, they do not carry the doggie umbrella, I must have seen it in Sears. I can't remember, but as I recall, it is a harness that keeps the umbrella in place.
I know that some dogs don't like to get their feet wet. When biking on the towpath over they years I have seen pups with raingear on (like a slick) and also boots. Never the umbrella..
Mine never mind.. in fact a few minutes ago as I was taking the rubbish out (1st load) they pushed out the door, in fairly heavy rain, and off they went after somehting - I suspect a possum, but then they will chase after other morning treasures