We define new deformable families of vertex operator algebras
$\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ associated to a large set of
S-duality operations in four-dimensional supersymmetric gauge theory. They are
defined as algebras of protected operators for two-dimensional supersymmetric
junctions which interpolate between a Dirichlet boundary condition and its
S-duality image. The $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOAs are
equipped with two $\mathfrak{g}$ affine vertex subalgebras whose levels are
related by the S-duality operation. They compose accordingly under a natural
convolution operation and can be used to define an action of the S-duality
operations on a certain space of VOAs equipped with a $\mathfrak{g}$ affine
vertex subalgebra. We give a self-contained definition of the S-duality action
on that space of VOAs. The space of conformal blocks (in the derived sense,
i.e. chiral homology) for $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ is
expected to play an important role in a broad generalization of the quantum
Geometric Langlands program. Namely, we expect the S-duality action on VOAs to
extend to an action on the corresponding spaces of conformal blocks. This
action should coincide with and generalize the usual quantum Geometric
Langlands correspondence. The strategy we use to define the
$\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOAs is of broader applicability and
leads to many new results and conjectures about deformable families of VOAs.