Two-neutrino double electron capture is a rare nuclear decay where two electrons are simultaneously captured from the atomic shell. For $^{124}\mathrm{Xe}$ this process has not yet been observed and its detection would provide a new reference for nuclear matrix element calculations. We have conducted a search for two-neutrino double electron capture from the $K$ shell of $^{124}\mathrm{Xe}$ using 7636 $\mathrm{kg}\phantom{\rule{0.16em}{0ex}}\mathrm{d}$ of data from the XENON100 dark matter detector. Using a Bayesian analysis we observed no significant excess above background, leading to a lower 90% credibility limit on the half-life ${T}_{1/2}g6.5\ifmmode\times\else\texttimes\fi{}{10}^{20}$ yr. We have also evaluated the sensitivity of the XENON1T experiment, which is currently being commissioned, and found a sensitivity of ${T}_{1/2}g6.1\ifmmode\times\else\texttimes\fi{}{10}^{22}$ yr after an exposure of 2 $\mathrm{t}\phantom{\rule{0.16em}{0ex}}\mathrm{yr}$.
