I was late wearing a bra although I needed one. It was because my mother insisted I should be properly fitted for my first bra, but she was too ill to take me bra shopping for several years. My first bra was a B cup, which was too big to have left unsupported for so long. And yes, girls at school were really mean and nasty about it. In the same way, girls who developed large and early were treated as morally bankrupt and promiscuous.
My advice - don't make her wear one if she doesn't want to. It has to be her choice. BUT - I would suggest to her that some sort of support would be useful, especially when she's doing kid stuff like climbing trees, running, jumping or anything really active. A sports bra would be good but I think it's too much for her emotionally at this stage. ANY shirt which provides hidden support would be a great idea - you've got summer coming in, I spent most of summer wearing a swimsuit top as a singlet top over a skirt. You can always throw a longer sleeved shirt over the top, or take off the loose shirt if it's really hot weather.
A singlet top with a shelf bra built in is maybe a good option - they can be tricky to pull on, with everything so tight, but once it's on it really disguises boobs. It also flattens them a bit so she won't look so big. She might like that. Here's a link which should show what I mean - http://www.flexonline.com.au/cgi-bin/site-editor.pl/8/-ecommerce?action=item;item=BW0028;eid=571977
Whether she likes it or not, her body is betraying her (in her mind). Denial will not make her boobs shrink. Ancient Greek women athletes would wear a cloth band over their chests to flatten everything and hold things steady so sport didn't cause pain. Anything you can persuade her to try that will have this same effect should be easier for her, as long as she can think of it as NOT a bra, but simply support for looser flesh. A white singlet top with built-in hidden shelf bra should make undressing in the gym at school a bit easier - she can leave the singlet top on under her sports uniform.
The other problem is BOYS. She is probably terrified at the thought of them finding her attractive and wants to look flatter. So LET her look flatter if she wants. The issue for her right now is comfort, not looks. If she can find something which provides support without making her chest look more obvious this should make her happy. Try and think like she does, with a deeper level of aware and concerned mummy.
Give her want she wants to keep her safe and comfortable. Work with her on this if she will let you. Make it clear YOUR aim is her physical and emotional comfort.