You should not blame yourself. Never blame yourself. If it is because she is bullied in school, it's the bullies you should blame.
The thing you can do, as I can see it, is merely presenting these things you heard from the doctor to her (if she hadn't already heard them) in a very calm way. Not accusing her of anything, and never force her to eat (well, obviously she does eat but throws it up). She knows that she should be eating, common knowledge is that bodies malfunction without nourishment, but I think that she feels a bit like I do myself (I'm disgusted by food and the thought of eating it).
Have you tried to talk to her? I mean, -really- tried?
Let her tell you her story. Ask her what the reason is that she throws up what she's eaten.
Not;
"But dear Emily, why? WHY are you doing this?! Tell me!"
But more like;
"Hey sweety, how are you feeling? Have you eaten anything today? No? Why not? Care to tell me?"
Don't push her into telling you. Let her tell you when she can. If you push her, she'll feel more guilty and maybe ashamed (the reactions are very unique from person to person, I'm only referring to how I would have reacted).
Try and be her friend in this situation, not so very much her mother. If she refuses to talk to you, have her see a psychologist or someone in the likening of one.
I hope this can be somewhat helpful. Let me know if you need someone to talk to.
//E