Do my family members need to reapply for their Non-O visas when I switch to a Non-B visa?

Doubtful you have a Non-B VISA and instead it's more likely you're on an "extension of stay" <-look at the stamp in your passport.

Your "piggy back" family members aren't on Non-O visas either they'd be on an extension of stay that matches yours

Which means IF you are leaving that employer you'd cancel ALL the extensions, exit/re-enter the country and everyone would start over from scratch. You with a new Non-B and them with new Non-O's based on you being employed here

Because your family is "piggy backing" on your extension when you change yours, theirs has to be changed too.