Can vomit get into the nasal and respiratory passages?

Not normally.
Vomiting causes reflex closure of the vocal cords, respiratory arrest and closure of the nasopharyngeal and epiglottic passages, which prevents inadvertent aspiration of vomit into the nasal and respiratory passages. However, when the patient loses consciousness or vomits too rapidly, the neural reflexes are not well modulated and therefore there is a risk of aspiration by mistake.