  In this note, we consider the twisted Yangians $\text{Y}(\mathfrak{g}_N)$
associated with the orthogonal and symplectic Lie algebras
$\mathfrak{g}_N=\mathfrak{o}_N,\mathfrak{sp}_N$. First, we introduce a certain
subalgebra $\text{A}_c(\mathfrak{g}_N)$ of the double Yangian for
$\mathfrak{gl}_N$ at the level $c\in\mathbb{C}$, which contains the centrally
extended $\text{Y}(\mathfrak{g}_N)$ at the level $c$ as well as its vacuum
module $\mathcal{M}_c(\mathfrak{g}_N)$. Next, we employ its structure to
construct examples of quasi modules for the quantum affine vertex algebra
$\mathcal{V}_c(\mathfrak{gl}_N)$ associated with the Yang $R$-matrix. Finally,
we use the description of the center of $\mathcal{V}_c(\mathfrak{gl}_N)$ to
obtain explicit formulae for families of central elements for a certain
completion of $\text{A}_c(\mathfrak{g}_N)$ and invariants of
$\mathcal{M}_c(\mathfrak{g}_N)$.
