We demonstrate the existence of a new type of zero energy state associated with vacancies in multilayer graphene that has a finite amplitude over the layer with a vacancy and adjacent layers, and the peculiarity of being quasilocalized in the former and totally delocalized in the adjacent ones. In a bilayer, when a gap is induced in the system by applying a perpendicular electric field, these states become truly localized with a normalizable wave function. A transition from a localized to an extended state can be tuned by the external gate for experimentally accessible values of parameters.