I am not a mom, just a kid who has been adopted and been failed by both sets of parents. I am also trans. I just say this so you can make your own decisions on how much weight you want to put on my advise.
In my opinion, others are right. At 13, spending all of their time in their room is not healthy (and as someone who was thirteen just 8 years ago, I wouldn't consider it normal either). Please get her help, at least a therapist.
She asked you for a binder. That's huge, I was far too afraid to ask my parents for a binder, that means there is trust (or desperation) there. I would look into getting her one from GC2B and before you ignore this please hear my reasoning. These are the safest binders out there, abd if she binds correctly, there should be no risks. Some women even wear binders simply because they prefer how they look with them. You can say this to her when you get her one. "Liking how you look in a binder doesn't mean you have to be a boy, girls can wear them too" this is important because if she isnt trans, you also don't want her to feel like she has to be in order to wear one. (my roommate, a woman wears one, she's not trans, just likes how it looks). Trust me, if she wants to bind and you won't let her get a binder, she very well might anyway, and it won't be safe. Bandages, ripped up cloth, even rope, old bras, many things can be used and were used to bind before safe options were available. Cheap binders online look like bras but bind so tight they crack ribs. Trust me, it's better to buy her one and making sure she's using it safely (not more that 8 hours a day, no sleeping or swimming etc) then to find out at her next doc appointment that she has bruised ribs and damaged tissue that would have been avoided had she been about to bind with a good quality binder.
Remember kids today are very different. We are encouraged to think for ourselves and see the world our own way much younger. So don't shove her in a box and force her to stay under your wing, behind your fence until you say she's old enough, much better to let her on a leash and guide her.