csv-table命令を使う際、$ と " の間の半角スペースが入っていたら、それを取り除くと上手くいきます。
.. csv-table:: :header: "", " $\bm{S_{0}}$ ", " $\bm{S_{1}}$ "," $\bm{S_{2}}$ "," $\bm{S_{3}}$ ", " $\bm{S_{4}}$ " " $\bm{S_{0}}$ ", " $S_{0}$ ", " $S_{1}$ "," $S_{2}$ "," $S_{3}$ ", " $S_{4}$ " " $\bm{S_{1}}$ ", " $S_{1}$ ", " $S_{2}$ "," $S_{3}$ "," $S_{4}$ ", " $S_{0}$ "
↓
.. csv-table:: :header: "", "$\bm{S_{0}}$", "$\bm{S_{1}}$","$\bm{S_{2}}$","$\bm{S_{3}}$", "$\bm{S_{4}}$" "$\bm{S_{0}}$", "$S_{0}$", "$S_{1}$","$S_{2}$","$S_{3}$", "$S_{4}$" "$\bm{S_{1}}$", "$S_{1}$", "$S_{2}$","$S_{3}$","$S_{4}$", "$S_{0}$"