We present results from a search for X-ray/gamma-ray counterparts of gravitational-wave (GW) candidates from the third observing run (O3) of the LIGO–Virgo–KAGRA network using the Swift Burst Alert Telescope (Swift-BAT). The search includes 636 GW candidates received with low latency, 86 of which have been confirmed by the offline analysis and included in the third cumulative Gravitational-Wave Transient Catalogs (GWTC-3). Targeted searches were carried out on the entire GW sample using the maximum-likelihood Non-imaging Transient Reconstruction and Temporal Search pipeline on the BAT data made available via the GUANO infrastructure. We do not detect any significant electromagnetic emission that is temporally and spatially coincident with any of the GW candidates. We report flux upper limits in the 15–350 keV band as a function of sky position for all the catalog candidates. For GW candidates where the Swift-BAT false alarm rate is less than 10−3 Hz, we compute the GW–BAT joint false alarm rate. Finally, the derived Swift-BAT upper limits are used to infer constraints on the putative electromagnetic emission associated with binary black hole mergers.