In this paper we apply the penalty function method to the vector variational inequality problem, in order to transform a constrained problem, referred to as the original problem, into a sequence of simpler, unconstrained problems, referred to as the penalized problems. We show that any limit point of a sequence of solutions of the penalized problems is a solution of the original problem. Moreover, under certain assumptions on the feasible region $D$ and the function $\mathbf{F}$, we can show that every penalized problem has a solution, and that a sequence of solutions of the penalized problems always has at least one limit point. As far as we know, this is the first study on solving a vector variational inequality problem using the penalty function method.