1. about any two organizations
The student affairs office at your school probably has the authority to decide whether or not the school will recognize the chapter. If the school's regulations allow, that office can probably decide to rely on the opinions of organizations that are already on campus before it will recognize a newcomer. The lack of school recognition does not necessarily prevent a group from colonizing, but many things can be tied to school recognition, including membership on interorganizational councils, the right to participate as a GLO in school-sponsored events, and even the right to live in large-group housing (because towns and cities often zone so that a fraternity can only occupy a house if it is recognized by the school).
As a matter of corporate law, whether or not a chapter exists depends on the charter granted by the parent organization. The school cannot dictate that the chapter does not exist. This is why you see some chapters that continue after the school has ceased to recognize them; as long as the (inter)national organization recognizes them, they exist. But the school doesn't have to let the organization participate in its events, and others can also tie rights and privileges to school recognition.
In practicality, though, many organizations won't try to colonize if they can't get approval from the school. If the school wants to delegate that decision to the already-existing system, it can probably do that.
2. about these organizations in particular
Is there something in particular about these two organizations that would give the school a reason to handle this matter this way? Or is there something that raises special cause for alarm? I don't know anything about either of them. Are they somehow related?
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