Stumbled on this again, I should make my comment an official answer.

This is true for any proper $J\subseteq S$. Suppose $g\in W$ fixes every coset of $W_J$, so $g$ lies in the intersection of all conjugates of $W_J$. Any intersection of parabolic subgroups (meaning conjugates of standard parabolic subgroups) is a parabolic subgroup, so the intersection of all conjugates of $W_J$ is a proper normal parabolic subgroup of $W$. But $(W,S)$ is finite and irreducible, so the only proper normal parabolic subgroup of $W$ is $\{1\}$. Hence $g=1$, i.e., the action is faithful.

(As a remark, I don't think this used finiteness of $W$ (??), just irreducibility of $(W,S)$. The key is just that there are no proper non-trivial normal parabolic subgroups.)