The question as stated probably requires clarification. If

$X$ is a space, then the S-dual $D_+(X)$ (i.e., functions from $X_+$ to the sphere) is always an $E_\infty$-ring spectrum. In particular, it will also be an $A_\infty$-ring.

Perhaps what is being asked is whether $D_+(G)$ is an $A_\infty$-coalgebra when $G$ is a topological group. The answer is yes, because  $G$ is an $A_\infty$-space.

Hence, $D_+(G)$ is Hopf algebra up to homotopy in such a way that the multiplication is $E_\infty$ and the comultiplication is $A_\infty$.

So $D_+(G)$ should be something like a homotopy everything version of an algebra over the Hopf operad.

The question as stated probably requires clarification. If

$X$ is a space, then the S-dual $D_+(X)$ (i.e., functions from $X_+$ to the sphere) is always an $E_\infty$-ring spectrum. In particular, it will also be an $A_\infty$-ring.

Perhaps what is being asked is whether $D_+(G)$ is an $A_\infty$-coalgebra when $G$ is a topological group. The answer is yes, because  $G$ is an $A_\infty$-space.