  We construct a new class of quantum vertex algebras associated with the
normalized Yang $R$-matrix. They are obtained as Yangian deformations of
certain $\mathcal{S}$-commutative quantum vertex algebras and their
$\mathcal{S}$-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. In particular, we show that its fixed points
are closely related with Bethe subalgebras in the Yangian quantization of the
Poisson algebra $\mathcal{O}(\mathfrak{gl}_N((z^{-1})))$, which were recently
introduced by Krylov and Rybnikov. Finally, we extend this construction of
commutative families to the case of trigonometric $R$-matrix of type $A$.
