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

$h$-adic quantum vertex algebras associated with rational $R$-matrix in types $B$, $C$ and $D$