Yes, the statement is true, since the underlying topological space of an orbifold $O$ is a manifold $|O|,$ and the natural map from $O$ to $|O|$ induces a surjective homomorphism on fundamental groups. On the other hand, that map is identity on the boundary.

The statement is true when $\partial O$ is a surface without cone points, since the underlying topological space of an orientable 3-orbifold $O$ is a manifold $|O|,$ and the natural map from $O$ to $|O|$ induces a surjective homomorphism on fundamental groups. On the other hand, that map is identity on the boundary.