The title essentially says it all. Feel free to assume as many finite generation conditions as you want. For example, I'm pretty sure the Chevalley-Eilenberg complex $\simeq \wedge^\bullet \mathfrak g^*$ for a (semisimple?) Lie algebra $\mathfrak{g}$ gives an étale cover of the classifying space $BG$ of the corresponding Lie group $G$.
I basically want to know if I can define a stack by descent by étale covers from these non-connective semi-free CDGAs.
Yes it is somehow true. I think what you are referring to is called formal localization, and is kinda written in CPTVV in a algebraic fashion and in Pridham's work in a more geometric fashion I believe (see the link in Pridham's comment). We work over a field of zero characteristic. I think to state things properly, one has to use (quasifree ?) mixed graded algebras instead of non-connective cdgas.
Let me try to explain how I understand formal localization from the CPTVV perspective (It is also explained in CPTVV in a short paragraph in the introduction).
Firstable, let me introduce mixed graded cdgas and derived foliations (as defined by Toën and Vezzosi here).
A mixed graded cdga $A$ is the data of a family of dgmodules $(A(n))_{n\in \mathbb{Z}}$ together with :
An affine derived foliation is the data of a graded mixed cdga $A$ such that :
Both are equipped with projective the model structure where weak equivalences and fibrations are checked gradedwise.
There is a slight difference with the definition of Toën and Vezzosi where they require as well $A(1)[-1]$ to be connective, I think because of some crystals issues (it behaves better if it is connective)). In some sense, a derived foliation is a quasi-free mixed graded cdga generated in weight $1$ and $0$.
For instance, if $\mathfrak{g}$ is a perfect dgLie algebroid acting on $\operatorname{Spec}(A)$, its Chevalley-Eilenberg complex has a structure of mixed graded cdga (where the mixed structure is given by the Chevalley differential and the grading is the symmetric power grading), and is actually a derived foliation. I think that conversely, a derived foliation gives a dgLie algebroid over $\operatorname{Spec}(A)$. I personnaly see a derived foliation intuitively as the data of an affine derived scheme and a dgLie algebroid acting on it.
Now, an important fact in deformation theory is that you can actually classify formal thickenings of $\operatorname{Spec}(A)$. A theorem of Nuiten (here) together with some of the work of Gaitsgory and Rozenblyum (A study in derived algebraic geometry available here) (See the survey of Grivaux and Calaque that summarizes what I am talking about well) states precisely that if $A$ is eventually coconnective, then there is an equivalence of $\infty$-categories between dgLie alegbroids over $\operatorname{Spec}(A)$ and formal thickenings of $\operatorname{Spec}(A)$, if you restrict your thickenings to be perfects, then it become an equivalence between derived foliations over $\operatorname{Spec}(A)$ and formal thickenings. The functor giving the dgLie algebroid is actually the functor that takes a formal thickening $\operatorname{Spec}(A) \rightarrow X$ and gives the relative tangent complex $\mathbb{T}_{\operatorname{Spec}(A)/X}$. The associated Chevalley-Eilenberg complex will be a derived foliation that can be seen as the relative de Rham complex $\mathbf{DR}(\operatorname{Spec}(A)/X)$.
For a derived stack $X$, its de Rham stack is the derived stack $X_{\operatorname{dR}} : A \mapsto X(H^0(A)^{\operatorname{red}})$. You can check that if $X$ is Artin, $X_{\operatorname{dR}}$ has a cotangent complex that is $0$. There is a natural map $X \rightarrow X_{\operatorname{dR}}$ that is close to being $X \rightarrow *$. A proposition in CPTVV tells us that this natural map is "formally affine" (Assume $X$ is a locally finitely presented Artin stack) :
If $\operatorname{Spec}(A) \rightarrow X_{\operatorname{dR}}$ is a point of the de Rham stack, it gives us a map $\operatorname{Spec}(H^0(A)^{\operatorname{red}}) \rightarrow X$, and the fiber of $X \rightarrow X_{\operatorname{dR}}$ along $\operatorname{Spec}(A) \rightarrow X_{\operatorname{dR}}$ is the formal neighborhood of the graph map $\operatorname{Spec}(H^0(A)^{\operatorname{red}}) \rightarrow \operatorname{Spec}(A)\times X$.
But because it is a formal thickening of the spectrum of a cdga eventually coconnective, the fiber is fully determined by a derived foliation that will be the relative de Rham stack, something totally algebraic. So you can see the morphism $X \rightarrow X_{\operatorname{dR}}$ as something that can be algebraicly encoded. This gives us a presheaf of mixed graded cdgas on $\mathbf{dAff}/X_{\operatorname{dR}}$ that CPTVV denotes $\mathbb{D}_{X/X_{\operatorname{dR}}}$.
Now, we have to see this presheaf over a presheaf of mixed graded cdgas representing $X_{\operatorname{dR}}$, so one has to embed connective cdgas into mixed graded cdgas, the right way to do this is to send $A$ to $\mathbf{DR}(A/A^{\operatorname{red}})$. Thus we get the presheaf denoted by $\mathbb{D}_{X_{\operatorname{dR}}}$ in CPTVV, and $\mathbb{D}_{X/X_{\operatorname{dR}}}$ is actually a $\mathbb{D}_{X_{\operatorname{dR}}}$-module.
Those two presheaves don't satisfy the etale descent but I believe they should satisfy some modified formally etale descent.
Now, if you want to define something using this descent, concretely, you can define something algebraically on these sheaves, and check that it coincides in the affine case, if your data is functorial for formally etale morphism it should work. For the reference, that is exactly what CPTVV does to define shifted Poisson structures on derived Artin stacks, because Poisson structures are actually not functorial in smooth morphisms.
Also, you mentionned that the Chevalley-Eilenberg of the Lie algebra of a Lie group $G$ provides a cover of $\mathcal{B}G$. I think it will instead provide a cover of the formal neighborhood of $* \rightarrow \mathcal{B}G$ instead of the whole $\mathcal{B}G$. For the whole $\mathcal{B}G$ one has to consider the presheaf of all formal neighborhoods like I did for $X$ over its de Rham stack I think. Maybe it will work in differential geometry, if you assume that $G$ is simply connected (but it amounts to say that $G$ is totally determined by its Lie algebra).
