A survey of 40,000 people found that 90%+ of all Americans have premarital sex. So statistically, even if your kids wait the chances are good that their partner didn't. And HPV is sneaky: it is entirely possible that your child's partner could have it and not know it and one of your kids could end up infected even if they both wait.
That said, why do your views on abstinence matter? If you don't have any problem with the vaccine from a health standpoint, shouldn't you want your kids to be as safe as possible even if they make decisions that you disagree with? There have been so many studies done that show that abstinence only education only delays the onset of sexual activity by about six months and the kids who wait that extra half year then have less safe sex with a higher rate of STI infection and pregnancy. Give your kids all of the information, protect them to the best of your ability (access to condoms, birth control, etc.), and then trust that you raised them right and that they are good kids who will make you proud.
On topic of the original article: I did a quick search on Google Scholar. There doesn't seem to be anything from any reliable sources that would indicate that the article is scientifically sound or really has any scientific basis at all. Here's some good information about premature ovarian failure, though:
This site provides a good discussion of the claim as well (although the author has some pretty strong opinions on the reasoning behind attacks on the drug):
These are the parts I found most relevant:
"[The authors of the article OP linked] point out in the introduction that premature ovarian failure has an estimated incidence of 10/100,000 person-years between the ages of 15 and 29. Other sources suggest that the incidence of such ovarian dysfunction might be as high as 1 in 1,000 before the age of 40. The authors also cite sources that find that the cause of ovarian failure before the age of 40 ďremains unknown in up to 90% of cases.Ē In other words, in the vast majority of cases premature ovarian failure is idiopathic. We donít know what caused it.
[emphasis added] In the cases where a cause can be identified, potential causes are several and include autoimmune disorders..., genetics, chemotherapy, hysterectomy..., thyroid dysfunction..., Turner syndrome, inadequate gonadotropin secretion or action, and viral infections.
...And when did this girl receive her anti-HPV vaccine? According to the case report, she received doses in May and August of 2008. Now letís go back to the case report. This girl started to have irregular menses in early 2009, more than five months after her last dose of Gardasil, and then she didnít stop menstruating until a year later, in early 2010."