  We report the $p+p$ and $p+d$ differential cross sections measured in the
SeaQuest experiment for $J/\psi$ and $\psi\left(2S\right)$ production at 120
GeV beam energy covering the forward $x$-Feynman ($x_F$) range of $0.5 < x_F
<0.9$. The measured cross sections are in good agreement with theoretical
calculations based on the nonrelativistic QCD (NRQCD) using the long-distance
matrix elements deduced from a recent global analysis of proton- and
pion-induced charmonium production data. The $\sigma_{\psi\left(2S\right)} /
\sigma_{J/\psi}$ cross section ratios are found to increase as $x_F$ increases,
indicating that the $q \bar{q}$ annihilation process has larger contributions
in the $\psi\left(2S\right)$ production than the $J/\psi$ production. The
$\sigma_{pd}/2\sigma_{pp}$ cross section ratios are observed to be
significantly different for the Drell-Yan process and $J/\psi$ production,
reflecting their different production mechanisms. We find that the
$\sigma_{pd}/2\sigma_{pp}$ ratios for $J/\psi$ production at the forward $x_F$
region are sensitive to the $\bar{d}/ \bar{u}$ flavor asymmetry of the proton
sea, analogous to the Drell-Yan process. The transverse momentum ($p_T$)
distributions for $J/\psi$ and $\psi\left(2S\right)$ production are also
presented and compared with data collected at higher center-of-mass energies.
