No, Stack Overflow does not provide this type of service / functionality.
You could look at commercial offerings such as AnswerHub. I have no affiliation with them, but my company does use a customized version of theirs for our own support / Q & A and it works by and large the same way this site does.
They may be able to provide a substantial educational discount; I have no insight into that, but you'd have to contact them directly to get reasonable estimates anyway.
http://answerhub.com/qa/questions/952/does-answerhub-have-special-pricing-for-non-profit.html
That is the one I have experience with; as pointed out in the comments, there are many othersthere are many others.