It is misogynistic to suggest that sex segregation in sports or locker rooms is necessary for women to succeed; it is also misogynistic and transphobic to exclude transgender and gender non-conforming women from the definition of women. Sex segregation in sports does not make sense. Rather than just assume that people assigned female at birth need separate leagues in every single sport to have meaningful competition, we should have separate leagues by ability, not by sex or gender. For those concerned about opportunity, I do not believe athletic scholarships should exist and I do not believe any leagues should exclude on the basis of sex or gender. 
The arguments for excluding trans and gender non-conforming women from locker rooms are even more ridiculous. It is inherently misogynistic to suggest that women are made unsafe by being in the presence of people with certain genitalia, or that to solve the problem of men attacking women, women need to be excluded from spaces where men exist. It's worth noting that this argument was made to justify excluding women from combat positions in the US military because of sexual assault in the US military; because of men's violence, people proposed women be excluded "for their own protection". There is no right to privacy from people only of a certain sex, gender, or genitalia, and pretending such a right exists is based in misogyny, transphobia, and nonbinary-phobia, not any genuine desire to stop gendered violence. 
It's worth noting that the people responsible for segregating bathrooms and sports by sex were rarely if ever working from foundations of egalitarianism, womanism, or feminism. Historical documentation shows that more often than not they were working from bases of misogyny, sexism, and transphobia. The Virginia Board of Health must not make the same mistakes, and should reject this petition and the insulting and wrong assumptions on which the petition is based.