Although the majority of Herbig Ae/Be (HAEBE) stars are likely multiple systems, relatively few multiples have been identified. We aim to create a statistically significant survey of HAEBE multiplicity as a function of stellar mass, to shed light on mixed-mass star formation. Two independent adaptive optics IR imaging surveys were conducted, between 1993 and 2007. A total of 143 stars were observed, including 126 HAEBE stars. These surveys are significantly more sensitive than previous imaging work and, combined with accurate distance and proper-motion data from GAIA Data Release 3, provide a broad survey of intermediate-mass multiplicity. We perform a statistical probability analysis to identify the likely physical companions, and analyze multiepoch observations for common proper motions. We find 250 stellar detections around 88 sample stars. Of these, 70 systems are likely multiple systems, and 42 companions around 29 systems are reported here for the first time. Correcting for completeness, we find a multiplicity fraction of 54%. Multiples are found randomly across all spectral types, but high-order multiples are more common among the hottest stars. Statistically likely companions are found at distances of up to several thousand astronomical units and {Delta}K magnitudes ranging from <0.1 to ~9, although companions with {Delta}K<1.5 are rare. Companions around Herbig Ae stars tend to be closer than those around Herbig Be stars. We confirm 24 proper-motion companions in 21 systems, including roughly half of the statistically very likely companions; however, many proper-motion results are inconclusive.