Yes, there is a normal force. The normal force is the name given to the manner in which the surface of any body will "push back" as a reaction to a force applied to that body. The rope does this.
Whether I consider myself standing on a floor or on a tight rope, I exert a force downward on that object. Newton's third law requires that the object exert an equal force upward on me - the normal force.
You should think of it in this manner - the vertical component of the tension in the rope will determine the strength of this normal force, and is its cause.
So, the tension in the rope causes a normal force to act on the walker. When we calculate the vertical component of the tension, we are simultaneously determining the normal force of the tight rope walker. We should not, however, view them as separate, individual forces. The rope would not be able to produce a normal force without having a tension applied to it.