Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there is a scheme $U_{\alpha}$ and a finite group $\Gamma_{\alpha}$ acting on $U_{\alpha}$ with the property that $\mathcal{X} \times_X X_{\alpha}$ is isomorphic to the stack theoretic quotient $[U_{\alpha}/ \Gamma_{\alpha}]$. Is something like this true for the Deligne-Mumford geometric derived stacks of Toen-Vezzosi? Vaguely, I understand the above statement to be true because of how etale maps are defined when your affine objects are actual schemes. (If $V$ is a scheme and $V \rightarrow \mathcal{X}$ is etale, the pullback of $V$ and the henselization of $X$ at a geometric point is finite over the henselization). If you have a much more general definition of etale, can this property be salvaged?

Yes, these type of local structure theorems also hold for derived stacks, even more general ones such as Theorem 1.2 in https://arxiv.org/abs/1504.06467. If $\mathcal{X}$ is a derived stack (+ adjectives), apply such a result to the classical truncation $t\mathcal{X}$ to get an etale covering $(X_\alpha^0 \to t\mathcal{X})_\alpha$ where $X^0_\alpha$ is the quotient of an affine by $\Gamma_\alpha$. The etale site is invariant under derived nilpotent thickenings such as $t\mathcal{X} \to \mathcal{X}$, so this lifts uniquely to an etale covering $X_\alpha \to \mathcal{X}$. The affine map $X_\alpha^0 \to B\Gamma_\alpha$ uniquely extends to an affine map $X_\alpha \to B\Gamma_\alpha$ (which means that $X_\alpha$ is the quotient of an affine by $\Gamma_\alpha$). This is because the obstructions to this extension are controlled by the cotangent complex $L_{B\Gamma_\alpha}$ which is 0 if $\Gamma_\alpha$ is a finite group (or more generally $L_{B\Gamma_\alpha}$ is concentrated in degree $1$ at most which is enough to conclude if $\Gamma_\alpha$ is linearly reductive as in the Alper-Hall-Rydh theorem).

One place such a structure theorem is used (without actually justifying..) is Corollary 5.1(3) in Toen's Inventiones paper on derived Azumaya algebras.