  Two-neutrino double electron capture is a process allowed in the Standard
Model of Particle Physics. This rare decay has been observed in $^{78}$Kr,
$^{130}$Ba and more recently in $^{124}$Xe. In this publication we report on
the search for this process in $^{124}$Xe and $^{126}$Xe using the full
exposure of the Large Underground Xenon (LUX) experiment, in a total of of
27769.5~kg-days. No evidence of a signal was observed, allowing us to set 90\%
C.L. lower limits for the half-lives of these decays of
$2.0\times10^{21}$~years for $^{124}$Xe and $1.9\times10^{21}$~years for
$^{126}$Xe.
