  A detailed account of the local atomic structure and disorder at 5~K across
the phase diagram of the high temperature superconductor
K$_x$Fe$_{2-y}$Se$_{2-z}$S$_z$ $(0 \leq z \leq 2)$ is obtained from neutron
total scattering and associated atomic pair distribution function (PDF)
approaches. Various model independent and model dependent aspects of the
analysis reveal a high level of structural complexity on the nanometer
length-scale. Evidence is found for considerable disorder in the $c$-axis
stacking of the FeSe$_{1-x}$S$_{x}$ slabs without observable signs of
turbostratic character of the disorder. In contrast to the related FeCh (Ch =
S, Se) type superconductors, substantial Fe-vacancies are present in
K$_x$Fe$_{2-y}$Se$_{2-z}$S$_z$, deemed detrimental for superconductivity when
ordered. Our study suggests that the distribution of vacancies significantly
modifies the iron-chalcogen bond-length distribution, in agreement with
observed evolution of the PDF signal. A crossover like transition is observed
at a composition of $z\approx1$, from a predominantly vacancy-disordered state
at the selenium end to a more vacancy-ordered (VO) phase closer to the sulfur
end of the phase diagram. The S-content dependent measures of the local
structure are found to exhibit distinct behavior on either side of this
crossover, correlating well with the evolution of the superconducting state to
that of a magnetic semiconductor towards the $z\approx2$ end. The behavior
reinforces the idea of the intimate relationship of correlated Fe-vacancies
order in the local structure and the emergent electronic properties.
