Yes. It's less likely, but still possible. As hero65 said, not all forms of flu are covered by the vaccine. Also, the vaccine is not 100% effective even for the forms it does cover. If the babysitter has the flu, she's probably in no shape to babysit anyway, but she shouldn't get near him until she's been symptom free without any medications for 24 hours. If he was already exposed, you'll have to wait and see. If you're worried about the rest of the house, pick up some masks for all of you to wear, and wash hands often. The flu incubation period is 1-4 days, so in 4 days you should know for sure.
yes he can still get sick because flu has many forms he got vaccinated against the most probable form in your area but still he is in danger so if he gets very sick when he catches the flu it's better to keep him away from her