A while after I had asked the above question back in October 2011 (on another forum, the question now having been imported here), the following article appeared, which claims to produce what I was wondering about:
based on
 Chungsheng James Yeh, Topics in superstring theory, PhD thesis, Berkeley 1993 (SPIRE)
Their introduction has a useful survey. First they recall the traditional way of going about superstring field theory:
The first attempt towards a field theory of superstrings was initiated by the work of Witten
 Edward Witten, Interacting field theory of open superstrings, Nuclear Physics B, Volume 276, Issue 2 (1986)
by seeking a ChernSimons like action for open superstrings similar to the one of open bosonic string field theory (Witten 86). The major obstacle compared to the bosonic string is the necessity of picture changing operators?. Indeed, the cubic superstring theory of (Witten 86a) turns out to be inconsistent due to singularities arising form the collision of picture changing operators
 C. Wendt, Scattering amplitudes and contact interactions in Witten’s superstring field theory, Nuclear Physics B, Volume 314, Issue 1.
In order to circumvent this problem, another approach was pursued which sets the string field into a different picture

C.R. Preitschopf, C.B. Thorn, S. Yost, Superstring field theory Nuclear Physics B, Volume 337, Issue 2.

I.Ya. Aref’eva, P.B. Medvedev, A.P. Zubarev, New representation for string field solves the consistency problem for open superstring field theory, Nuclear Physics B, Volume 341, Issue 2.
but upon including the Ramond sector, the modified superstring field theory suffers from similar inconsistencies
 M. Kroyter, Superstring field theory equivalence: Ramond sector, Journal of High Energy Physics, Volume 2009, Issue 10.
These two approaches are based on the small Hilbert space, the state space including the reparametrization ghosts and superghosts as they arise from gaugefixing. Upon bosonization of the superghosts, an additional zero mode arises which allows the formulation of a WZW like action for the NS sector of open superstring field theory
In contrast to bosonic string field theory, BV quantization of this theory is more intricate than simply relaxing the ghost number constraint for the fields of the classical action

Nathan Berkovits, Constrained BV description of string field theory, Journal of High Energy Physics, Volume 2012, Issue 3.

M. Kroyter, Y. Okawa, M. Schnabl, S. Torii, Barton Zwiebach, Open superstring eld theory I: gauge xing, ghost structure, and propagator, Journal of High Energy Physics, Volume 2012, Issue 3.
Finally, there is a formulation of open superstring field theory that differs from all other approaches in not fixing the picture of classical fields
 M. Kroyter, Superstring field theory in the democratic picture, Advances in Theoretical and Mathematical Physics, Volume 15, Number 3.
Then they come to the "geometric" approch which I was asking about whether it has been generalized from the bosonic to the superstring:
On the other hand, the construction of bosonic closed string field theory (Zwiebach 92) takes its origin in the moduli space of closed Riemann surfaces. Vertices represent a subspace of the moduli space, such that the moduli space decomposes uniquely into vertices and graphs,and do not apriori require a background. Graphs are constructed from the vertices by sewing together punctures along prescribed local coordinates around the punctures. But an assignment of local coordinates around the punctures, globally on the moduli space, is possible only up to rotations. This fact implies the level matching condition and via gauge invariance also the (b_0^)constraint.
In an almost unnoticed work (Yeh), the geometric approach developed in bosonic closed string field theory, as described in the previous paragraph, has been generalized to the context of superstring field theory. NeveuSchwarz punctures behave quite similar to punctures in the bosonic case, but a Ramond puncture describes a divisor on a super Riemann surface rather than a point. As a consequence, local coordinates around Ramond punctures, globally defined over super moduli space, can be fixed only up to rotations and translation in the Ramond divisor.
A given background provides forms on super moduli space

A. Belopolsky, New Geometrical Approach to Superstrings, hepth/9703183.

L. AlvarezGaume, P. Nelson, C. Gomez, G. Sierra, C. Vafa, Fermionic strings in the operator formalism, Nuclear Physics B, Volume 311, Issue 2.
in the sense of geometric integration theory on supermanifolds, and in particular the geometric meaning of picture changing operators has been clarified
Integrating along an odd direction in moduli space inevitably generates a picture changing operator. Thus, the ambiguity of defining local coordinates around Ramond punctures produces a picture changing operator associated with the vector field generating translations in the Ramond divisor. The bpz inner product plus the additional insertions originating from the sewing define the symplectic form relevant for BV quantization. As in the bosonic case, we require that the symplectic form has to be nondegenerate, but the fact that the picture changing operator present in the Ramond sector has a nontrivial kernel, forces to impose additional restrictions besides the level matching and b_0^ = 0 constraint on the state space. The purpose of (JurcoMuenster 13) is to describe the construction of type II superstring field theory in the geometric approach.
To the extent that this is correct, it means that also closed superstring field theory has an action functional that is a ChernSimonstype term for a (super) Lie nalgebra (for n = infinity, a "strong homotopy Lie algebra"), that's equation (4.12) of (JurcoMuenster 13).