  Let $(X,\omega)$ be a symplectic orbifold which is locally like the quotient
of a $\mathbb{Z}_2$ action on $\reals^n$. Let $A^{((\hbar))}_X$ be a
deformation quantization of $X$ constructed via the standard Fedosov method
with characteristic class being $\omega$. In this paper, we construct a
universal deformation of the algebra $A^{((\hbar))}_X$ parametrized by
codimension 2 components of the associated inertia orbifold $\widetilde{X}$.
This partially confirms a conjecture of Dolgushev and Etingof in the case of
$\mathbb{Z}_2$ orbifolds. To do so, we generalize the interpretation of Moyal
star-product as a composition of symbol of pseudodifferential operators in the
case where partial derivatives are replaced with Dunkl operators. The
star-products we obtain can be seen as globalizations of symplectic reflection
algebras.
