Answer

Declaring property ownerless can be done only by the owner or by a sufficiently important religious court (some say: the singular leading authority of the generation). I do not think that the local halakhic authority can declare property ownerless, because otherwise every religious court could annul a marriage by declaring the money ownerless.
It may perhaps be said that the seven leading men of the town act as representatives of the residents, and therefore their authority to declare property ownerless is grounded in the community’s authority to declare ownerless the property of an individual within it—like a person declaring his own property ownerless.
It is hard to say that the yeshiva staff counts as the seven leading men of the town. They were not elected and do not represent the public. So this is closer to the local halakhic authority than to the seven leading men of the town.
But in my opinion they can remove the belongings from the area under their control even without declaring them ownerless, after prior warning to the students and their families.