Naively we may expect: Diff(M/G) = Diff(M) // G, where Diff(M) // G = Diff(M)^G /I, where "I" is two-sided ideal in Diff(M)^G of operators which act by zero on invariant functions.
This is example of quantization commute with reduction ideology, since Diff(N) - quantization of T*N.
My questions is what is the state of art ? I would expect it is known to be true if action of M on G is free, M - smooth oriented manifold. If yes what is the reference ?
There is paper by B. Fedosov:
Non-Abelian Reduction in Deformation Quantization, Lett. Math. Phys. 1998 Volume 43, Number 2, 137-154,
As far as I understand his result should imply the positive answer for the compact group G. But I am not sure about the details, may be he assume some compactness, or some other assumptions...
If any one would be so kind to send me this paper, it would be very kind of him.