  Given an action $\varphi$ of of inverse semigroup $S$ on a ring $A$ (with
domain of $\varphi(s)$ denoted by $D_{s^*}$) we show that if the ideals $D_e$,
with $e$ an idempotent, are unital, then the skew inverse semigroup ring
$A\rtimes S$ can be realized as the convolution algebra of an ample groupoid
with coefficients in a sheaf of rings. Conversely, we show that the convolution
algebra of an ample groupoid with coefficients in a sheaf of rings is
isomorphic to a skew inverse semigroup ring of this sort. We recover known
results in the literature for Steinberg algebras over a field as special cases.
