Yep, there's no doubt that for us who love river fishing there is no still water can compare.
I'm afraid I'm not familiar with South Yorkshire and it's surrounds, so I'm sorry I'm not able to help with an answer to your question. I know there are many Yorkshire people on this board though, so I'm sure one of them at least will have some ideas.
Anyway, stick with it, good luck and hopefully you'll find some nice quiet swim that's nice and easy to get to.
With the disability what is it that gives you problems, by the way? It might be good for the respondents to know.
Is it that you can't walk too far, or can't get over stiles or up and down sloping banks or what?
I know of one spot on the Don in Sheffield, quite a nice steady stretch with a little more depth than average. From memory it is just above the Stevenson road bridge. The river is a few feet below the bank and railed off. However sections of the railings can be opened to give fishing access. I have seen wheelchair anglers there in the past, I don't know any contacts to obtain keys though. There may be notices up at the site.
Secondly, it is also possible to access the river further into Sheffield centre from Warren Street. There is an entrance marked 'Five Weirs Walk' take this and turn right at the river, there is a large wooden platform above the weir with low railings. Before they dug the weir out this was an excellent spot for chub, perch, grayling, dace and the odd trout. This is 'free'.