If a horse pulls a wagon from rest and the wagon pulls back equally does the wagon move?

Yes, the wagon will move if the horse pulls it. While the horse and wagon exert equal and opposite forces on each other (according to Newton's third law), the horse is capable of generating enough force to overcome the inertia of the wagon. As a result, the horse's force will cause the wagon to accelerate forward, assuming the ground provides sufficient traction for the horse.