  Although the study of X-ray binaries has led to major breakthroughs in
high-energy astrophysics, their circumbinary environment at scales of
$\sim$100--10,000 astronomical units has not been thoroughly investigated. In
this paper, we undertake a novel and exploratory study by employing direct and
high-contrast imaging techniques on a sample of X-ray binaries, using adaptive
optics and the vortex coronagraph on Keck/NIRC2. High-contrast imaging opens up
the possibility to search for exoplanets, brown dwarfs, circumbinary companion
stars, and protoplanetary disks in these extreme systems. Here, we present the
first near-infrared high-contrast images of 13 high-mass X-ray binaries located
within $\sim$2--3 kpc. The key results of this campaign involve the discovery
of several candidate circumbinary companions ranging from sub-stellar (brown
dwarf) to stellar masses. By conducting an analysis based on galactic
population models, we discriminate sources that are likely
background/foreground stars and isolate those that have a high probability
($\gtrsim 60 - 99\%$) of being gravitationally bound to the X-ray binary. This
publication seeks to establish a preliminary catalog for future analyses of
proper motion and subsequent observations. With our preliminary results, we
calculate the first estimate of the companion frequency and the multiplicity
frequency for X-ray binaries: $\approx$0.6 and 1.8 $\pm$ 0.9 respectively,
considering only the sources that are most likely bound to the X-ray binary. In
addition to extending our comprehension of how brown dwarfs and stars can form
and survive in such extreme systems, our study opens a new window to our
understanding of the formation of X-ray binaries.
