  Planet formation is often considered in the context of one circumstellar disk
around one star. Yet stellar binary systems are ubiquitous, and thus a
substantial fraction of all potential planets must form and evolve in more
complex, dynamical environments. We present the results of a five-year
astrometric monitoring campaign studying 45 binary star systems that host
Kepler planet candidates. The planet-forming environments in these systems
would have literally been shaped by the binary orbits that persist to the
present day. Crucially, the mutual inclinations of star-planet orbits can only
be addressed by a statistical sample. We describe in detail our sample
selection and Keck/NIRC2 laser guide star adaptive optics observations
collected from 2012 to 2017. We measure orbital arcs, with a typical accuracy
of ~0.1 mas/yr, that test whether the binary orbits tend to be aligned with the
edge-on transiting planet orbits. We rule out randomly-distributed binary
orbits at 4.7$\sigma$, and we show that low mutual inclinations are required to
explain the observed orbital arcs. If the stellar orbits have a field
binary-like eccentricity distribution, then the best match to our observed
orbital arcs is a distribution of mutual inclinations ranging from 0-30
degrees. We discuss the implications of such widespread planet-binary alignment
in the theoretical context of planet formation and circumstellar disk
evolution.
