Hi OP, thank you for being so caring, but I am afraid that neither your YABU or YANBU quite cover my choice. I would really like at least several hours to think about this, but I will give you my initial thoughts.
a) How would you feel if "A's" mother took such an equivalent decision for your DD, as you are thinking of doing for hers.
b) Have you looked on the Internet to see if chest binding is even safe or effective for what "A" wants to do? My first thought is that it could damage her breast tissue, maybe even leading to breast cancer in a worse case scenario.
c) You have known "A's" parents for 11 years and think that they are loving (they most probably are, but no-one really knows how someone else would react to any given situation if they had never even discussed it either them before). I think that her mum does need to know her plans, and if you could persuade her to tell her mum - with you present if she would feel happier/safer, then I think that would be the best outcome.
I presume that their secondary school has some kind of pastoral care, if she refuses to talk to her mum could you persuade her to talk to an appropriate person at the school?
If she refuses both of those options then if you know who her GP is, or at least what surgery she goes to, I think you would have to write a letter to them, labelling it private, to only be opened by a GP. Please give them your name and address (email address?), because even though they cannot discuss "A's" case with you, they might want to ask you further questions, and to even verify that you are who you say you are. The responsibility will then lie with the GP about whether they tell the parents or not. I know that at 15 years old they still should, but many would probably take all sorts of things into consideration, some of which you may not even be aware of.
So to sum up OP, no I definitely would not order anything to bind her chests (that sounds little better to me than when Chinese girls/women had their feet bound), but I would tell someone, as suggested above, about her desire to do so. I don't like such young people taking hormone suppressants, but that is my personal viewpoint, as to me I would want them to devope naturally, but once they are 18 (or even 16 with the right expert advice) encourage them to start dressing as the sex they feel they want to be, like having appropriate hair styles, etc, anything really that is not a permanent change, as I feel that the years of puberty, and early adulthood are a maelstrom of emotions, and that until their hormones have had a chance to calm down, and they have at least a few life experiences under their belts, it is not a good time to make irreparable decisions.
Having said all that, it can be very hard not to support some desperate youngsters when you hear them talk, and when they say that ever since they were 2 or 3 years old they have hated all the gender influences aimed at them, which in this case could have been - dolls, clothes, hairstyles, over protectiveness that maybe a brother didn't experience, etc. - these poor children would need exceptional expert help, and maybe to go on suppressing or stimulating hormones when they are still children? If that is the case I can only hope and pray that the experts are there to help them, and with no underlying agendas of their own.
The reason I cannot say YANBU is because you qualify it with saying you won't buy the binders without the parents permission. I am absolutely certain that binders are the wrong way to go, so I wouldn't buy them for someone whether their parents agreed or not.