To me, the illness history makes her age irrelevant. I'd apply the same thinking to someone of 16, 19 or 30+ to be honest. It's not about how mature she is, it's about how healthy she is.
I've had anorexia since I was 15. I'm now 35. Many of my closest friends have or have had it too. One friend who is suffering particularly badly right now couldn't be trusted to get herself to the corner shop and back independently right now - she's 32. Twice the age of the girl in question. Age doesn't count in illness.
I'd need to know more about your daughter's health to answer properly.
If she's a recovered anorexic who still needs to be a bit careful/follow regular patterns then yes, I'd let her go.
If she's in recovery then I might work with her and/or her treatment team to find a safe way or a partial compromise to allow her to have a bit of fun and independence but not jeopardise her safety or recovery.
If she's actively engaging in behaviours then no, I wouldn't let her go. Both for her sake and for her friends. Anorectics are often a nightmare to be around. Deceitful, manipulative, selfish, irrational, reckless, reclusive, mean, obsessive - any or all of those things plus others. If your brain is starving then your reaction to something as simple as getting lost, falling out with a friend or losing your phone can be hugely disproportionate and frightening. Your daughter has already has problems with suicide attempts. Could she handle an everyday sort of holiday problem?