So the short answer is that there is not such a model structure. The difficulty arises in trying to show that the class of weak equivalences has all of the necessary properties; in particular, even two-of-three does not hold for the naive definition. The first difficulty arises even before that: on ordinary simplicial sets we can arrange for a model of every set that is "minimal" on the $\pi_0$-level, meaning that every connected component has exactly one $0$-simplex. In simplicial commutative monoids we can no longer do this. However, we could assume that in order to be a weak equivalence we need to be a $\pi_*$-isomorphism when choosing any (coherently chosen) basepoints.
For the purposes of our discussion we are going to assume that $\pi_*$-equivalences use the model $S^n = \Delta^n/\partial\Delta^n$. (This is the model that most closely mimicks the boundary maps in the Dold-Kan correspondence.) Now let $X = S^2$, and let $Y$ be $S^2$ with an extra $0$-simplex connected by a $1$-simplex to the original basepoint. (So it looks like a balloon on a string.) We define a map $X\rightarrow Y$ to be the inclusion of $S^2$ in the obvious manner, and a map $Y\rightarrow X$ to be collapsing the extra $1$-simplex back down. Then the composition of these two maps is the identity on $X$, so obviously a weak equivalence. The map $X\rightarrow Y$ is also a weak equivalence, because adding the "string" can't add any new homotopy groups to $X$. However, the map $Y\rightarrow X$ is not a weak equivalence, as $\pi_2Y$ based at the extra point is a one-point set but $\pi_2X$ at its image is a two-point set.
The problem arose because in order to show that $\pi_*$ was invariant of basepoint in the usual Kan complex model we needed to be able to "pull back" simplices along paths in the simplicial set, which used the Kan condition. The new model does not have such a condition, and thus we can't necessarily pull things back.
Another observation along these lines. Take any connected simplicial set $X$, and let $Y$ be $X$ with a "string" added to it at any basepoint. Then $*\rightarrow Y$ (including into the new point) is a weak equivalence, and $X\rightarrow Y$ (including into itself) is a weak equivalence. Thus in the homotopy category, $X$ is isomorphic to a point (and thus the homotopy category is just the category of sets) ... which is presumably not desired.
-- The Bourbon seminar
I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the notion of "direct sum group" in Wagoner's paper (which unfortunately seems to be a little mangled).
Let $M$ be a homotopy commutative topological monoid with $\pi_0(M)=\mathbb{N}$. Choose a point $1 \in M$ in the correct component and let $n \in M$ be the $n$-fold product of 1 with itself, and define $G_n = \pi_1(M,n)$. The monoid structure defines homomorphisms
$$\mu_{n,m} : G_n \times G_m \longrightarrow G_{n+m}$$
which satisfy the obvious associativity condition. Let $\tau : G_n \times G_m \to G_m \times G_n$ be the flip, and
$$\mu_{m,n} \circ \tau : G_n \times G_m \longrightarrow G_{n+m}$$
be the opposite multiplication. Homotopy commutativity of the monoid $M$ not not ensure that these two multiplications are equal, but it ensures that there exists an element $c_{n,m} \in G_{n+m}$ such that
$$c_{n,m}^{-1} \cdot \mu_{n,m}(-) \cdot c_{n,m} = \mu_{m,n} \circ \tau(-).$$
Let $G_\infty$ be the direct limit of the system $\cdots \to G_n \overset{\mu_{n,1}(-,e)}\to G_{n+1} \overset{\mu_{n+1,1}(-,e)}\to G_{n+2} \to \cdots$.
Theorem: the derived subgroup of $G_\infty$ is perfect.
Proof: Let $a, b \in G_n$ and consider $[a,b] \in G'_\infty$. Let me write $a \otimes b$ for $\mu_{n,m}(a, b)$ when $a \in G_n$ and $b \in G_m$, for ease of notation, and $e_n$ for the unit of $G_n$.
In the direct limit we identify $a$ with $a \otimes e_n$ and $b$ with $b \otimes e_n$, and we have
$$b \otimes e_n = c_{n,n}^{-1} (e_n \otimes b) c_{n,n}$$
so $b \otimes e_n = [c_{n,n}^{-1}, (e_n \otimes b)] (e_n \otimes b)$. Thus
$$[a \otimes e_n, b \otimes e_n] = [a \otimes e_n, [c_{n,n}^{-1}, (e_n \otimes b)] (e_n \otimes b)]$$
and because $e_n \otimes b$ commutes with $a \otimes e_n$ this simplifies to
$$[a \otimes e_n, [c_{n,n}^{-1}, (e_n \otimes b)]].$$
We now identify this with
$$[a \otimes e_{3n}, [c_{n,n}^{-1}, (e_n \otimes b)] \otimes e_{2n}]$$
and note that $a \otimes e_{3n} = c_{2n,2n}^{-1}(e_{2n} \otimes a \otimes e_{n})c_{2n,2n} = [c_{2n,2n}^{-1}, (e_{2n} \otimes a \otimes e_{n})]\cdot (e_{2n} \otimes a \otimes e_{n})$.
Again, as $(e_{2n} \otimes a \otimes e_{n})$ commutes with $[c_{n,n}^{-1}, (e_n \otimes b)] \otimes e_{2n}$ the whole thing becomes
$$[a,b]=[[c_{2n,2n}^{-1}, (e_{2n} \otimes a \otimes e_{n})], [c_{n,n}^{-1}, (e_n \otimes b)] \otimes e_{2n}],$$
a commutator of commutators.
Best Answer
In view of the references to my Memoir, Classifying spaces and fibrations, in other answers, I guess I should answer too. The requested answer is implicit but not quite explicit there. Fix a grouplike topological monoid $G$. Maybe assume for simplicity that its identity element is a nondegenerate basepoint (no loss of generality by 9.3). Define a $G$-torsor to be a right $G$-space $X$ such that the $G$-map $G\longrightarrow X$ that sends $g$ to $xg$ is a weak equivalence for $x\in X$. Let $\mathcal{G}$ be the category of $G$-torsors and maps of $G$-spaces between them. The Memoir defines a $\mathcal{G}$-fibration in terms of the $\mathcal{G}$-CHP, which is equivalent to having a $\mathcal{G}$-lifting function. Cary asks whether that notion is equivalent to an a priori weaker notion. The answer depends on what ``equivalence'' means. The Memoir insists on $\mathcal{G}$-fibrations, but it defines an equivalence (6.1) to be a $\mathcal{G}$-map over the base space, where a $\mathcal{G}$-map only has to be a map of $G$-torsors on fibers. (More precisely, it takes the equivalence relation generated by such maps.) Using $\mathcal{G}$-fibrations allows one often to replace that notion of equivalence by the nicer one of $\mathcal{G}$-fiber homotopy equivalence. But with the equivalence relation as given, any $\mathcal{G}$-map that is a quasifibration is equivalent to a $\mathcal{G}$-fibration, by the $\Gamma$-construction in Section 5. Therefore the classification theorem remains true allowing all $\mathcal{G}$-spaces that are quasifibrations, which of course includes Cary's preferred notion of a $G$-fibration.