I think you are talking not just about regular "high tops" but the fashionable super-high tops that look like a boot (they go to just below the knee) but are a lace-up sneaker, right?
I would definitely not advise wearing those for gym at school. Our gym teachers and coach make kids change out of any form of boot and into regular sneakers (the gym keeps a stash of extra sneakers) for gym. These are basically boots, not real athletic shoes. Get them for her just for fun if you want but don't expect them to double as supportive athletic shoes.
You say she'd have to wear them daily but she also can't try them on. Not a great combination of factors there. She may love the look but quickly find it irritating to have something laced up over her whole calf all day, every day. Just something to think about.I would get her one pair of good, quality sneakers for gym days and playground visits and a cheap, knockoff pair of these high-top-boots for fun, rather than an expensive name brand version.