I'm not sure I understand the question, we have a graded commutative algebra --- how can you deform it to an $L(\infty)$ algebra? However, one can simply regard poly-vector fields as a Lie algebra and ask what are the $L(\infty)$ deformations of that. I'm not sure this has a nice answer.

Tamarkin's approach to proving Kontsevich formality is to upgrade this idea and realize that poly-vector fields has a $G(\infty)$ structure, which is essentially the GCA structure together with the Lie structure. And then one should study the deformations of that $G(\infty)$ algebra. I'm not sure if this has a clean answer for a general variety or manifold either.

However, this deformation theory is indeed trivial for say $\mathbb{C}[x_1,...,x_n]$. From there, there is a local to global patching argument of the local formality isomorphisms. This is the part of the proof that uses formal geometry.

This idea works in fact for proving the analogue of Kontsevich formality for $\mathbb{C}[x_1,...,x_n]$ thought of as an $E_n$ algebra for any $n$,
"generalized Kontsevich formality". I don't know if this argument patches nicely and the known proof of generalized Kontsevich formality proceeds via some reduction scheme.