>Is there really a university IT department out there that's going to prevent students and professors from using an offsite, external website?

Yep. They'll quote policy and if that doesn't work pull out the FERPA card. But it isn't insurmountable. The ongoing push towards more unified assessment systems by the accreditation bodies means that many universities are purchasing third party systems that are hosted off-campus, and that seems to be opening up some avenues.