  We show a non-vanishing result for the averages of the derivatives of
$L$-functions associated with the orthogonal basis of the space of
vector-valued cusp forms of weight $k\in \frac12 \mathbb{Z}$ on the full group
in the critical strip. We also show the existence of at least one basis element
whose $L$-function does not vanish under certain conditions. As an application,
we generalize our result to Kohnen's plus space and prove an analogous result
for Jacobi forms.
