  We prove a vector-valued version of Mui\'c's integral non-vanishing criterion
for Poincar\'e series on the upper half-plane $ \mathcal H $. Moreover, we give
an accompanying result on the construction of vector-valued modular forms in
the form of Poincar\'e series. As an application of these results, we construct
and study the non-vanishing of the classical and elliptic vector-valued
Poincar\'e series.
