Which preposition should follow facilitate?
None of the sentences are, in my view, grammatical.
But you could say "The staff should facilitate access to on-line support for students". It is not the "students" who are being facilitated, but "access for students". You can facilitate a process, an event, an action, or a result etc., but you cannot, in my opinion facilitate a person.
But "enable" works differently. You could say "The staff should enable students to access on-line support". It is a more versatile word, because you could also "enable access for students".