  This work investigates the implications of very special relativity (VSR) on
the calculation of vacuum polarization for fermions in the presence of Maxwell
and Kalb-Ramond gauge fields in four-dimensional spacetime. We derive the
$SIM(2)$-covariant gauge theory associated with an Abelian antisymmetric
2-tensor and its corresponding field strength. We demonstrate that the free
VSR-Kalb-Ramond electrodynamics is equivalent to a massive scalar field with a
single polarization. Furthermore, we determine an explicit expression for the
effective action involving Maxwell and Kalb-Ramond fields due to fermionic
vacuum polarization at one-loop order. The quantum corrections generate
divergences free of nonlocal terms only in the VSR-Maxwell sector. At the same
time, we observe UV/IR mixing divergences due to the entanglement of
VSR-nonlocal effects with quantum higher-derivative terms for the Kalb-Ramond
field. However, in the lower energy limit, the effective action can be
renormalized like in the Lorentz invariant case.
