Good question. I think the answer is yes.

The unnamed spectral sequence is usually referred to as the *isotropy spectral sequence*. For a group $G$ acting on $X$ and an abelian group $A$ of coefficients as in your question, it has $E_2$-page given by the sheaf cohomology $H^p(X/G; \mathcal{H}^q)$ of the orbit space, and converges under favourable circumstances to the cohomology $H^*(EG\times_G X;A)$ of the homotopy orbit space. If $X$ is a regular $G$-complex, then the sheaf $\mathcal{H}^q$ of coefficients takes the value $H^q(G_\sigma;A)$ over a simplex $[\sigma]$ of the orbit space $X/G$. Summarising, the spectral sequence goes from cohomology of the orbit space with coefficients in the cohomology of the isotropy subgroups, to the equivariant cohomology.

This can be identified with the *Leray spectral sequence* of the map $EG\times_G X\to X/G$ from the homotopy orbit space to the (genuine) orbit space, given by projecting $EG$ to a point. This is not a fibration in general, but one still gets a spectral sequence, at the expense of replacing cohomology with local coefficients by sheaf cohomology. A decent reference for the Leray spectral sequence of map is the book of Bott and Tu.

Now, the Leray spectral sequence of a map works just as well for generalized cohomology theories. The details appear in the thesis of Richard Cain,

<cite authors="Cain, R.N.">_Cain, R.N._, [**The Leray spectral sequence of a mapping for generalized cohomology**](http://dx.doi.org/10.1002/cpa.3160240106), Commun. Pure Appl. Math. 24, 53-70 (1971). [ZBL0205.53002](https://zbmath.org/?q=an:0205.53002).</cite>

So for a generalized cohomology theory $F^*$ and a regular $G$-complex $X$, you should get a spectral sequence with $E_2$-page $H^p(X/G; \mathcal{F}^q)$ converging to $F^*(EG\times_G X)$, where $\mathcal{F}^q$ takes the value $F^q(G_\sigma)$ over a simplex $[\sigma]$ of $X/G$.