In the reaction ${\mathrm{ClO}}_3^{-}\left(aq\right)+5{\mathrm{Cl}}^{-}\left(aq\right)+6{\mathrm{H}}^{+}\left(aq\right)⟶3{\mathrm{Cl}}_2\left(g\right)+3{\mathrm{H}}_2\mathrm{O}\left(l\right)$ the oxidizing and reducing agents are, respectively :

This reaction described in problem is a comproportionating reaction. In this reaction oxidizing agents is ${\mathrm{ClO}}_3^{-}\left(aq\right)$ and reducing agent is ${\mathrm{Cl}}^{-}\left(aq\right)$.

Thus, the correct option is B.