  We study groupoid actions on left cancellative small categories and their
associated Zappa-Sz\'ep products. We show that certain left cancellative small
categories with nice length functions can be seen as Zappa-Sz\'ep products. We
compute the associated tight groupoids, characterizing important properties of
them, like being Hausdorff, effective and minimal. Finally, we determine
amenability of the tight groupoid under mild, reasonable hypotheses.
