As Safari Steve mentioned, people under 18 are not allowed to rent the
ECVS in the parks. I have no personal experience with the outside rental places, but I have been told they also will not rent ecvs for use by someone under 18.
Here are the offsite rental places that DIS posters report renting from:
Care Medical:
http://www.caremedicalequipment.com/
Phone (407) 856-2273 Toll Free U.S and Canada (800) 741-2282
Walker Mobility:
1-888-726-6837
www.walkermobility.com
RANDY'S Mobility is in Kissimmee 407-892-4777
http://randysmobility.com/
Colonial Medical
http://www.colonialmed.com/about_cms.html
(800)747-0246
http://www.scootaround.com/ Not much feedback. Several people who reported they had no problems.
Randy's requires that someone be there for delivery and pick up. The rest will deliver to Bell Services and leave it for you.
I can attest that people using wheelchairs don't get to the head of the lines and sometimes have to wait longer to board (our record is an extra 40 minutes wait for the Safari at AK when people in the "regular" line had less than a 5 minute wait.)
AK and the Studio have almost 100% wheelchair accessible lines. Epcot and MK have some lines that are not totally accessible, but that's because they are older parks and there was no way to make them totally accessible. For those rides that don't have accessible lines, when the park is not busy, the CM may sometimes take the wheelchair user in right away. But that is because it is more convenient right then for the CM to handle the wheelchair party, not as a "perk" or advantage to the wheelchair user. For some rides, wheelchair users need to board at the exit. In most cases, they will wait in the regular line until just before the boarding area, and then they are pulled off to the exit. Even though they disappear from the line, they usually will be waiting at the exit to board approximately the same amount of time they would have waited to board in the regular line. When we ride Buzz Lightyear, we often see the people who were right ahead of us in line as they exit while we are still waiting at the exit to get on.
Someone mentioned a Guest Assistance Card. You can go to Guest Services in any of the parks and request a Guest Assistance Card (we call them GACs to avoid typing so much). In general, people who use wheelchairs or ecvs don't need and won't be given a GAC because just having the wheelchair/
ecv alerts the CM that they need an accessible entrance. If that meets your needs, you don't need a GAC. But some people need a quieter place to wait, a place out of the sun, a place away from other people or to be able to bring a stroller into line and have it treated the same as a wheelchair. The GAC is a tool to alert CMs of that sort of invisible need. It is not meant to (and usually won't) shorten or eliminate your wait in line.
Here's some pixie dust for your daughter's speedy recovery *~*~*~*~*~*~*~*~*~
Probably several weeks. He said what everyone else is saying about getting her a wheelchair. He mentioned the air cast and said we could do that but she might just be more hot and miserable at WDW with it. 
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