  We study quantum current algebra $\textrm{A}(\overline{R})$ associated with
the rational $R$-matrix of $\mathfrak{gl}_N$ 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 $\mathcal{V}_c(\overline{R})$ for the
algebra $\textrm{A}(\overline{R})$ possesses a quantum vertex algebra structure
for any complex number $c$. We prove that any module for the quantum vertex
algebra $\mathcal{V}_c(\overline{R})$ is naturally equipped with a structure of
restricted $\textrm{A}(\overline{R})$-module of level $c$ and vice versa.
