Hi Freddie,
Unfortunately there is no special fare for disabled persons or children on the JR. However most stations and trains come equipped facilities to make your travel as smooth as possible, think about priority seats, specially equipped toilets and extra large elevators and gates. You can also ask the JR Staff for any help you may require. Of course you can buy the JR Pass for you and your child if you enter Japan under a tourist Visa and children between 6-11 always pay half price.
I would also recommend reading this post over at Danny Choo's about traveling with a wheelchair.
I hope this helps!
-Daniel