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 $zapprox1$, 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 $zapprox2$ end. The behavior reinforces the idea of the intimate relationship of correlated Fe-vacancies order in the local structure and the emergent electronic properties.
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