  Following the idea of Aganagic--Okounkov \cite{AOelliptic}, we study vertex
functions for hypertoric varieties, defined by $K$-theoretic counting of
quasimaps from $\mathbb{P}^1$. We prove the 3d mirror symmetry statement that
the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are
equivalent to each other, with K\"ahler and equivariant parameters exchanged,
and the opposite choice of polarization. Vertex functions of a 3d mirror pair,
as solutions to the $q$-difference equations, satisfying particular asymptotic
conditions, are related by the elliptic stable envelopes. Various notions of
quantum $K$-theory for hypertoric varieties are also discussed.
