She'll probably hate you for it if you tell her, but if it's the only way to get her to get help, then you may have to. You could go and tell the doctor what you read on your own, but you may not have a say at all if she's over 18. But then again, if she's still in college, some doctors may bend the rules for you. (They did when my Mom got insistent and I was in college and having "problems".)
The important thing is, if she needs specialized treatment, to get it now, if you possibly can. When she's out of college and on her own and still out of control she may come to a point where she wants treatment but cannot get it to save her life. Treatment for eating disorders is so expensive, and if she doesn't get a job with good insurance benefits, she won't have the option of getting help later.
I guess . . . my Mom having read my journal when I was in college and was hurting myself . . . I got over it. I mean, you feel violated for awhile, but as you get older you really become a different person so you don't mind as much anymore. But, I mean, I have never been more angry with her in my life.
It also may be that your daughter is desperate for someone to see how out of control she feels and it would be a relief to finally get that acknowledgement. She may not want it from you (I didn't), but . . .
It's just better to get the truth on the table.
The thing about bulimia is that you feel so ashamed of it - that's probably why she's lying to you. I wish I could tell you exactly what to say. Just recognize that she feels ashamed, and if you spend too much time telling her how horrible what she's doing is, then she'll just feel worse. If it were me, I'd want someone to say:
"I know you feel out of control right now, but you just need a little bit of help, that's all. You can get back on track. We'll get you that help, and you won't have to worry anymore, because you're going to beat this, and you'll have the whole rest of your life ahead of you COMPLETELY FREE of this eating disorder."