Hi and welcome.
Because your DD is only 6, I would suggest you keep to your plan to
rent a stroller for her. The rental wheelchairs at the park are adult sized and might not be comfortable for a small child. You might want to look into renting from one of the offsite companies so you have it for going other places like DD. Some of the resorts have long walks between your room and the main area (like food court and pool area). People often don't realize how much walking there is from one place to the other. At Epcot, the distance around the WS is about 1 mile! The distances are similar at other parks.
People have posted about renting equipment for kids (regular strollers, special needs strollers and pediatric wheelchairs) from
http://www.all-about-kids.com/main.htm
For someone who uses a wheelchair and doesn't have any other special needs, a GAC isn't necessary. The wheelchair is the cue that some accomidationis needed. You can get a GAC that allows you to use a stroller as a wheelchair. That would mean you get to take the stroller into line and other places, like the buildings in Epcot where strollers are not normally allowed. You don't need anything special to get a GAC. Just go to Guest Services in any of the parks with your DD and tell them what problems you forsee.
In order to keep her walking some of the time, you might want to have her in the stroller between rides and have her walk in line. One caution to that though: many of the boarding areas have moving belts for boarding. She may find that is a little difficult for her to handle. The special needs/wheelchair boarding area for many rides is in the exit area. That allows a little extra time for boarding. Sometimes the CM can slow the belt. If you have her walk in the lines, it may not be evident to the CMs that she needs extra assistance until it's time to board, which is too late to get to the exit area. If she had a GAC, she could be routed to the exit more easily. You can get a Guidebook for Guests with Disabilities that gives some basic (very basic) information about boarding. There is one for each park and you can get them at Guest Services.
PS. We have tipped over a stroller when DD was getting out of it when there was too much weight on the back of the stroller (backpacks, etc). Your child is a counterweight to what everever else is in the stroller. When she gets out, the stroller will tip over because the weight is no longer in balance. That can happen with a wheelchair too, but most wheelchairs have antitip bars at the back to avoid that problem.