You might be interested in the paper by Mike Mandell titled : Topological Andre-Quillen Cohomology and $E_{\infty}$ Andre-Quillen Cohomology. I am pretty sure his statements do not require a
characteristic 0 assumption. For example, Theorem 1.3 on page 6 might be of interest.

Sorry, I guess I forgot to mention some things. I was initially pointed to this by Shipley's paper on $H\mathbb{Z}$ algebras and DGAs. The point is that when you want to remove the characteristic zero assumption you need to remember the $E_{\infty}$ algebra structure. In characteristic zero, it has to be essentially unique. This is related to the fact that the (co)homology of the symmetric groups is usually torsion. I remember Akhil Mathew asking a question related to this a while back. Anyways, the characteristic result follows. Although, it was known prior to this, see Clark Barwick's comment.

Rational Homotopy Theory, p. 223. See also Bousfield-Gugenheim. (This critically uses characteristic zero, of course; in characteristic $p$, the homology of a commutative simplicial algebra admits a divided power structure.) $\endgroup$