Two-neutrino double electron capture is a rare nuclear decay where two electrons are simultaneously captured from the atomic shell. For $^{124}$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 $^{124}$Xe using 7636 kg$cdot$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 $T_{1/2}>6.5times10^{20}$ yr. We also evaluated the sensitivity of the XENON1T experiment, which is currently being commissioned, and find a sensitivity of $T_{1/2}>6.1times10^{22}$ yr after an exposure of 2 t$cdot$yr.
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