As far as actual teaching goes, Vicki is the only one who we see doing any. IIRC, we are treated to scenes where they discuss Maine history, geography, etc. It's never stated clearly what her qualifications actually are, but given what we're told of her age and her background, I don't think she has any college classes behind her. It's implied that she did some teaching at the orphanage (similar to what Rachel must have done, albeit with a more secular bent and happily).
Others have posted here that most likely private governesses did not need to have extensive credentials. I don't know how true that is, but I do know that private school teachers usually don't need to be certified. I'm assuming that the original idea was that David would get private tutoring for a few years until his behavioral problems got under control and then they'd be shipping him to some prep school.
It makes less sense in later years when Barnabas hires Maggie, who has no experience teaching (at least not that the writers bothered to give her), and the children are much older.
