We present the construction of a new class of quantum vertex algebras associated with a normalized Yang R-matrix. They are obtained as Yangian deformations of certain S-commutative quantum vertex algebras and their S-locality takes the form of a single RTT-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. These results were obtained jointly with Slaven Kozic. If time allows, we will discuss a recent generalization of these results to the case of the type A trigonometric R-matrix. These results were obtained jointly with Marijana Butorac and Slaven Kozic.