An umbrella holder could work. What might prove more user friendly is a hood type device that could fold forward when not in use and pop over head with a press of a button when needed, much like the hood on a baby carriage.

I can imagine any number of mounting pieces already available in similar forms, from drum sets/percussion racks, plumbing braces, patio furnishings, and such.

An interesting idea might be an umbrella/parasol mounted in one leg of a walker, made retractable, wind-resistant, and easy-to-use.

The only problem I see with this is the kindof catch 22 it creates, large amounts of rain are very frequently accompanied by large amounts of wind. Let loose in a gust of wind an umbrella itself can be quite hazardous, then immagine a cart coming with it. That could be solved by using a breakaway style clamp(ie a certain amount of force in any direction could break the umbrella free) but even then you probably would have even more risk of a flying umbrella than if a human were holding it. I think all in all in boils down to a liability thing, not too many companies would be able justify the risk involved. not a bad idea tho