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 Image 1 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 $q\overline{q}$ 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 $\overline{d}/\overline{u}$ 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.