If you're talking about renting near WDW, chances are you won't be able to rent for a 15 y/o. The majority of vendors (including Disney) require that their powered mobility aids (scooters & powerchairs) only be used by people 18 or older. There is one offsite vendor who may rent scooters to older teens, but usually will only rent to teens who use one at home. If you're talking about purchasing, then, obviously, there aren't any age restrictions.
I'm not sure exactly what you mean by an "electric scooter chair". Do you mean an electric wheelchair with a joystick (also known as a powerchair)? In general a scooter type
ECV (with the bicycle-like handling) is easier to learn how to use. Unless she uses a powerchair at home, I wouldn't even try to use one for the first time renting at WDW. Again, I'm not sure if any vendors would rent to her at 15, anyway (most will only rent powerchairs to those who use them at home due to the difficulty with maneuvering and only to 18+). If you're talking about buying, it would be worth talking to her doctors and your insurance about who sells medical equipment near you and if you can get it covered by insurance. If you're going to spend the amount it costs to buy, she really should try before any money is paid.
If there are any big box stores or grocery stores near you that have scooters for patrons to use, it might be worth finding out if they'll let someone her age use one in the store. If they did, she could then try that to get used to how scooters move. The ones in stores are generally much slower and harder to maneuver than scooters you'd rent or buy so if she were to become comfortable with one at a store, she shouldn't have issues with one in other locations.
I was about her age when I developed mobility problems. Honestly, many fewer people stare at WDW than do in the "real world". There are so many people of a wide range of ages who use mobility aids at WDW that if people stared at them all, they'd never see Disney! Most are so wrapped into their world that you're more likely to not be seen by other guests (many don't see anything that's lower than their eye level) than to be stared at. Of the stares that do happen, a scooter or powerchair won't get any fewer stares than a manual wheelchair. I know it's harder to deal with mobility problems when you're a teenager because most teenagers already feel so out of place. One thing you may want to point out to her, though, is that the only people at WDW she's likely to see again are the members of your party.
She can always park a mobility aid by stroller parking and walk when she wants or needs to. It doesn't need to be in any photos. Since it sounds like she's okay walking without assistance at home, the only attractions that would be fully off limits are Tom Sawyer Island & Swiss Family Robinson Treehouse. (She can't use a mobility aid at all in accessing Tomorrowland Transit Authority and Peter Pan can't be slowed so she'd need to be able to get on with the normal amount of loading time.) The majority of rides, she wouldn't even need to use a different queue (for the few where she would, up to 5 or 6 other members of your party would be able to go with her).
If she absolutely refuses a wheelchair and tries to walk everything, she will be stuck walking and standing. There's no way for her to wait somewhere that she can sit and join the rest of you when you get to the front of a queue. Some of the queues are long distances that can't be cut down at all anyway (Soarin's queue is 1/4 mile just one way so to get to the ride and exit, it's 1/2 mile with no way to shorten it). Most people walk at minimum 4 miles a day. The average is 6 miles or more per day. Another thing to point out with the mobility aid is that it may help reduce her pain. I know that while I was fighting the idea of using a mobility aid, I'd limited my world down to very little because of pain and fatigue. Once I was willing to use a wheelchair, it was like an entire world had opened up (or re-opened) for me. I was older than she is by then, but I know now that if I'd had an appropriate mobility aid younger, I'd have been able to do far more of the "typical" teenage things.
I'd suggest actually posting this to the main disAbilities board (this is the community board more aimed at non-Disney topics and not everyone reads this one). In the stickies on top of the main board there's a FAQ about visiting WDW with a disability that has loads of good information. There are posters with lupus and other similar conditions who also post on the mail disAbilities board.