The CALorimetric Electron Telescope (CALET) on the International Space Station consists of a high-energy cosmic-ray CALorimeter (CAL) and a lower-energy CALET Gamma-ray Burst Monitor (CGBM). CAL is sensitive to electrons up to 20 TeV, cosmic-ray nuclei from Z = 1 through Z ∼ 40, and gamma rays over the range 1 GeV–10 TeV. CGBM observes gamma rays from 7 keV to 20 MeV. The combined CAL-CGBM instrument has conducted a search for gamma-ray bursts (GRBs) since 2015 October. We report here on the results of a search for X-ray/gamma-ray counterparts to gravitational-wave events reported during the LIGO/Virgo observing run O3. No events have been detected that pass all acceptance criteria. We describe the components, performance, and triggering algorithms of the CGBM—the two Hard X-ray Monitors consisting of LaBr 3(Ce) scintillators sensitive to 7 keV–1 MeV gamma rays and a Soft Gamma-ray Monitor BGO scintillator sensitive to 40 keV–20 MeV—and the high-energy CAL consisting of a charge detection module, imaging calorimeter, and the fully active total absorption calorimeter. The analysis procedure is described and upper limits to the time-averaged fluxes are presented.