I claim that for every $A_\infty$-space $A$, there is a canonical $A_\infty$-ring structure on $\Omega^\infty\Sigma^\infty_+A$.
First, $\Sigma^\infty_+$ from spaces to spectra is symmetric monoidal. So it sends an $A_\infty$-space $A$ to an $A_\infty$-algebra in spectra $\Sigma^\infty_+A$, that is an $A_\infty$-ring spectrum. The fact that $\Sigma^\infty_+$ is symmetric monoidal (at the model category/∞-category level) can be found in any modern book treating the smash product of spectra (e.g. it is proposition 4.7 in Elmendorf-May-Kriz-Mandell and corollary 188.8.131.52 in Lurie's Higher Algebra).
Secondly I claim that if $E$ is an $A_\infty$-ring spectrum, then $\Omega^\infty E$ has a canonical $A_\infty$-ring space structure. Exactly how this works will depend on your preferred definition of $A_\infty$-ring space, but it can be proven, e.g., with the same technique that May uses to prove the analogous statement for $E_\infty$-ring spaces (Corollary 7.5 in May's What precisely are $E_\infty$-ring spaces and $E_\infty$-ring spectra).
If all you care for is a construction of the $A_\infty$-structure on $GL_1(\Sigma^\infty_+A)$, I particularly like the approach in
Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, Charles Rezk An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology
where they identify $GL_1(R)$ with the automorphism group of $R$ as an $R$-module (and so it has an $A_\infty$-structure, since all automorphism groups in an ∞-category "trivially" do).