Yes!
Assume we are in characteristic $\neq2$. Then, your $\sigma$ acts as $\pm{\rm id}$ on a local generator of $\omega_X$. Since $X$ is an Enriques surface, we have $\omega_X^{\otimes2}\cong{\cal O}_X$.
We may thus find local generators $m_\alpha$ of $\omega_X$ on some open cover $U_\alpha$ of $X$, s.th. the $m_\alpha^{\otimes2}$ glue to the unique global section of $\omega_X^{\otimes 2}$. But then, $\sigma$ acts trivially on this global section (since it acts via multiplication by $\pm{\rm 1}$ on $m_\alpha$), i.e., on all global sections of $\omega_X^{\otimes2}$.
It is much less obvious, but still true, that automorphisms of order $3$ and $5$ also act trivially on global sections of $\omega_X^{\otimes2}$. Mukai and Ohashi exploit this in their recent analysis of automorphisms of Enriques surfaces.