Morality undermined by sexual notions of the importance of a good reputation
This chapter is all about people who look after their public reputations more than their souls. This is something Wollstonecraft finds in both genders, not just women.
Wollstonecraft is disgusted by how many women think that having a good reputation is the same thing as being a morally good person.
One of the reasons women focus so much on their reputation (especially their sexual reputation) is because in Wollstonecraft's world, there's no getting your reputation back once a person has "ruined" herself by having sex out of marriage. In this case, even a rumor can ruin a person's life forever, so it makes sense that women focus more on their reputations than anything else.
Wollstonecraft is willing to admit that reputation is usually a pretty good indication of a person's character, but she still thinks people focus on it too much instead of the person inside.
While talking about sexuality and reputation, Wollstonecraft argues that there should be a law forcing men to take care of the women they have seduced and had sex with.
It's total hypocrisy that men get to run around having all the sex they want while women are completely shunned for having sex before marriage. Nothing will ever improve for women unless men are willing to improve, too.