  We study properties of vertex (operator) algebras associated with 3d
H-twisted $\mathcal{N}=4$ supersymmetric gauge theories with a boundary. The
vertex operator algebras (VOAs) are defined by BRST cohomologies of the
currents with symplectic bosons, complex fermions, and bc-ghosts. We point out
that VOAs for 3d $\mathcal{N}=4$ abelian gauge theories are fermionic
extensions of VOAs associated with toric hyper-K\"{a}hler varieties. From this
relation, it follows that the VOA associated with the 3d mirror of $N$-flavors
$U(1)$ SQED is a fermionic extension of a $W$-algebra
$W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$. For $N=3$, we explicitly compute
the OPEs of elements in the BRST cohomology and find a new algebra which is a
fermionic extension of a Bershadsky-Polyakov algebra $W^{-2}(\mathfrak{sl}_3,
f_{\text{sub}})$. We also suggest an expression of the vacuum character of the
fermionic extension of $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$ predicted by
3d $\mathcal{N}=4$ mirror symmetry.
