  We introduce the $h$-adic quantum vertex algebras associated with the
rational $R$-matrix in types $B$, $C$ and $D$, thus generalizing the
Etingof--Kazhdan's construction in type $A$. Next, we construct the
algebraically independent generators of the center of the $h$-adic quantum
vertex algebra in type $B$ at the critical level, as well as the families of
central elements in types $C$ and $D$. Finally, as an application, we obtain
commutative subalgebras of the dual Yangian and the families of central
elements of the appropriately completed double Yangian at the critical level,
in types $B$, $C$ and $D$.
