  We introduce a variant of the $K$-theoretic quantized Coulomb branch
constructed by Braverman--Finkelberg--Nakajima, by application of a new virtual
intersection theory. In the abelian case, we define Verma modules for such
virtual Coulomb branches, and relate them to the moduli spaces of quasimaps
into the corresponding Higgs branches. The descendent vertex functions, defined
by $K$-theoretic quasimap invariants of the Higgs branch, can be realized as
the associated Whittaker functions. The quantum $q$-difference modules and
Bethe algebras (analogue of quantum $K$-theory rings) can then be described in
terms of the virtual Coulomb branch. As an application, we prove the
wall-crossing result for quantum $q$-difference modules under the variation of
GIT. Nonabelian cases are also treated via abelianization.
