In this paper we examine the well-posedness of evolutionary variational-hemivariational inequalities involving a constraint set and history-dependent operators. The strong and weak formulations of such inequalities are studied. First, the existence and uniqueness of solutions to both formulations are proved, and results on the dependence of solution on functional parameters are delivered. Next, the well-posedness is established for a general form of history-dependent variational-hemivariational inequalities with constraints by using a fixed point theorem. Finally, the results are applied to a dynamic frictional contact problem in viscoelasticity in which the contact is described by Signorini-type unilateral boundary condition with a nonmonotone Clarke’s relation.