  We prove that there is a positive proportion of $L$-functions associated to
cubic characters over $\mathbb{F}_q[T]$ that do not vanish at the critical
point $s=1/2$. This is achieved by computing the first mollified moment using
techniques previously developed by the authors in their work on the first
moment of cubic $L$-functions, and by obtaining a sharp upper bound for the
second mollified moment, building on work of Lester-Radziwill, which in turn
develops further ideas from the work of Soundararajan, Harper, and
Radziwill-Soundararajan. We work in the non-Kummer setting when $q \equiv
2\pmod{3}$, but our results could be translated into the Kummer setting when
$q\equiv 1\pmod{3}$ as well as into the number field case (assuming the
Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is
explicit, but extremely small, due to the fact that the implied constant in the
upper bound for the mollified second moment is very large.
