We study quantum current algebra A(R) associated with rational R-matrix and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan’s construction, the level c vacuum module Vc(R) for the algebra A(R) possesses a quantum vertex algebra structure for any complex number c. We prove that any module for the quantum vertex algebra Vc(R) is naturally equipped with a structure of restricted A(R)-module of level c and vice versa.