We started a survey with CHARA/MIRC-X and VLTI/GRAVITY to search for low-mass companions orbiting individual components of intermediate-mass binary systems. With the incredible precision of these instruments, we can detect astrometric "wobbles" from companions down to a few tens of microarcseconds. This allows us to detect any previously unseen triple systems in our list of binaries. We present the orbits of 12 companions around early F- to B-type binaries, 9 of which are new detections and 3 of which are first astrometric detections of known radial velocity (RV) companions. The masses of these newly detected components range from 0.45 to 1.3 M (circle dot). Our orbits constrain these systems to a high astrometric precision, with median residuals to the orbital fit of 20-50 mu as in most cases. For seven of these systems we include newly obtained RV data, which help us to identify the system configuration and to solve for masses of individual components in some cases. Although additional RV measurements are needed to break degeneracy in the mutual inclination, we find that the majority of these inner triples are not well aligned with the wide binary orbit. This hints that higher-mass triples are more misaligned compared to solar and lower-mass triples, though a thorough study of survey biases is needed. We show that the ARMADA survey is extremely successful at uncovering previously unseen companions in binaries. This method will be used in upcoming papers to constrain companion demographics in intermediate-mass binary systems down to the planetary-mass regime.