I believe the answer is yes for $C^\infty$ maps and actions of compact (not necessarily finite) Lie groups. I think it is due to  Poénaru and can be found in his book
*Singularités $C^\infty$ en présence de symétrie*
 Lecture Notes in Mathematics, Vol. 510. 

See also Lemma 6.6.1 in   *Dynamics and symmetry* by Michael Field.
(ICP Advanced Texts in Mathematics, 3. Imperial College Press, London, 2007. xiv+478 pp. ISBN: 978-1-86094-828-2)

(edit to reply to Brett's comment):
Poénaru's theorem is not holomorphic.  However,  I believe it should not be hard to mimic its proof to extract the holomorphic version.   I should note that I am not much of an expert on this area of mathematics and I know it more or less as a collection of black boxes.  My impression, however, is that in going from polynomial versions the results (which is classical invariant theory) to $C^\infty$ version the main difficulty is in dealing with smooth invariant functions that vanish to infinite order.  Going from polynomials to power series is not hard. And holomorphic maps from $V$ to $W$ are power series, aren't they?

Note also that in your example there is a big difference between complex $\mathbb Z/3$ invariant polynomials on $\mathbb C$ and real invariant polynomials on $\mathbb C$: $\mathbb C [\mathbb C]^{\mathbb Z/3}$ is generated by $z^3$ while $\mathbb R[\mathbb C]^{\mathbb Z/3}$ is generated by $Re(z^3), Im (z^3)$ and $|z|^2$.