There definitely are rooms without closets as one of my kids suitemates had one on Old Campus and he had one in his residential college. In those rooms they supplied an armoire. I don't know if they have rooms like that in Vanderbilt but I wouldn't assume that all the rooms have closets and they are just not shown on the floor plan.
BTW, his Old Campus room with a closet had at most 2.5 linear feet of hanging space with some type of 6" encased pipe running up through the center of the space for two to share. These dorms are not the place for someone who likes to surround themselves with a big wardrobe.