Yes, each resort has a number of wheelchairs available, for free, for any of their guests. They are first come, first served (can't reserve ahead of time) but we haven't had a problem getting them (although we go in the offseason). You can keep the chair for your entire stay, which saves on renting at the parks.
If the resort is out of chairs, (1) keep checking with them, as another may show, (2) they are available for rent at each theme park, but only for use at the park (but if you park-hop you can get a chair at the second park for free--only one rental to pay for per day), and (3) there are a number of services in the phone book who will deliver a rental chair to you if necessary.
I have not needed, but I expect that guest services at Downtown Disney may have chairs for use while shopping there as well.
Good luck, I'm sure grandma will have lots of fun.