I get very nervous around the word "required," but I have no problem otherwise. You need to get the vaccine before you begin having sexual contact in order for it to be effective--that's why it should be done so young. Not so that they can run right out and have sex, but because it's during a time when they shouldn't be having it, and when the right time does come they'll be protected from the start.
And, rough as it is to think about, not all sexual contact is consentual. I have been seriously considering getting it myself, even though I am not sexually active and probably won't be for a long time, simply because I belong to a social class that is particularly likely to be sexually assaulted.
Also, for the most part, I am strongly in favor of using natural consequences as a form of discipline--you refuse to wear a coat, you get cold; you break something, you pay for it. But, it is just not appropriate if the consequences are potentially fatal, the child is not able to fully appreciate them, and the parent has the ability to prevent them.
Sure, if nature is allowed to take it's course, the natural consequences of sneaking off to play in the pool alone is that you might drown. That's not a consequence I would "use" though. If I see it happening, I will prevent it.
Likewise, the natural consequence of unprotected sex is that you might contract STD's and, in this case, die of cancer many years down the line. That's another consequence I won't be using. Given the ability to prevent it, I will.
And besides all that, there's no reason the girl would have to know which vaccine they're getting, or the exact cause and nature of the disease.
Heck, I still don't really know what mumps and rubella are...except that I probably won't get them.
I see no reason why it shouldn't be treated the same way as Hep B (also an STD) vaccines--which my school required around this age. We actually got them in school, in the teachers' lounge. I don't think anybody was made more likely to run out and have sex, or get amature tattoos simply because they were vaccinated against a likely consequence.
Think about the vaccines you have right now:
Does having a Hep B vaccine compromise your own sense of sexual morality?
Does having a tetanus vaccine compromise the care you take with sharp, rusty objects?
Does having a safety net make a tightrope walker more likely to fall?