  Identifying the astrophysical sites of the $r$-process, one of the primary
mechanisms by which heavy elements are formed, is a key goal of modern
astrophysics. The discovery of the brightest gamma-ray burst of all time, GRB
221009A, at a relatively nearby redshift, presented the first opportunity to
spectroscopically test the idea that $r$-process elements are produced
following the collapse of rapidly rotating massive stars. Here we present
spectroscopic and photometric $\textit{James Webb Space Telescope}$ (JWST)
observations of GRB 221009A obtained $+168$ and $+170$ rest-frame days after
the initial gamma-ray trigger, and demonstrate they are well-described by a
supernova (SN) and power-law afterglow, with no evidence for an additional
component from $r$-process emission, and that the SN component strongly
resembles the near-infrared spectra of previous SNe, including SN 1998bw. We
further find that the SN associated with GRB 221009A is slightly fainter than
the expected brightness of SN 1998bw at this phase, concluding that the SN is
therefore not an unusual GRB-SN. We infer a nickel mass of $\approx0.09$
M$_{\odot}$, consistent with the lack of an obvious SN detection in the
early-time data. We find that the host galaxy of GRB 221009A has a very low
metallicity of $\approx0.12$ Z$_{\odot}$ and our resolved host spectrum shows
that GRB 221009A occurred in a unique environment in its host characterized by
strong H$_2$ emission lines consistent with recent star formation, which may
hint at environmental factors being responsible for its extreme energetics.
