The following is surely expressing whatever is in the core non-formal aspect of Joe Berner's answer (which is above my pay grade); it is offered as an alternative version of the same ideas.

Let $X$ be a normal noetherian scheme. We'll show the higher etale cohomology with coefficients in any flat $\mathbf{Z}$-module $M$ is torsion, and hence vanishes when $M$ is a $\mathbf{Q}$-vector space. Via the exact sequence $$0 \rightarrow M \rightarrow \mathbf{Q} \otimes_{\mathbf{Z}} M \rightarrow T \rightarrow 0$$ with $T$ visibly torsion, we just have to show the higher etale cohomology of the middle term vanishes; i.e., we are back to the original question of vanishing of higher etale cohomology with coefficients in a $\mathbf{Q}$-vector space $V$.

We can assume $X$ is connected, so it is irreducible (as $X$ is noetherian and normal); let $\eta$ be its unique generic point. For any quasi-compact etale $E \rightarrow X$, the scheme $E$ is normal and noetherian, so its connected components are all irreducible and as such have their unique generic point over $\eta$ (due to flatness of $E$ over $X$). Thus, $E \mapsto E_{\eta}$ does not lose any information about connected components, so for the natural map $j:\eta \rightarrow X$ we see that the natural map of etale sheaves $V_X \rightarrow j_{\ast}(V_{\eta})$ induces an isomorphism upon evaluating on any such $E$, whence it is an isomorphism of sheaves. Thus, our task is the same as proving the vanishing of the higher etale cohomology on $X$ of $j_{\ast}(V_{\eta})$.

Consider the Leray spectral sequence $$E_2^{n,m} = {\rm{H}}^n(X, {\rm{R}}^mj_{\ast}(V_{\eta})) \Rightarrow {\rm{H}}^{n+m}(\eta,V_{\eta}).$$ The sheaf ${\rm{R}}^m j_{\ast}(V_{\eta}$ is the etale sheafification of the presheaf on $X_{\rm{et}}$ given by $U \mapsto {\rm{H}}^m(U_{\eta},V)$ (with quasi-compact etale $U$ over $X$). The latter vanishes for $m>0$ since it is Galois cohomology (for the finitely many points of $U_{\eta}$) and $V$ is a $\mathbf{Q}$-vector space. Hence, $E_2^{n,m}=0$ for $m>0$, so $${\rm{H}}^n(X, j_{\ast}(V_{\eta})) \simeq {\rm{H}}^n(\eta,V_{\eta})$$ for all $n \ge 0$. But the right side vanishes for $n>0$ for the same Galois-cohomological reasons, so we are done.