I am amplifying the above comments. The theorem of Buch-Mihalcea also holds for stacks.

Let $k$ be a field of characteristic $0$. Let $Y$ be a Deligne-Mumford stack that is finite type over $k$ and that is normal: there exists a surjective, étale morphism $h:\widetilde{Y}\to Y$ with $\widetilde{Y}$ a normal scheme. Let
$f:X\to Y$ be a proper, surjective morphism of Deligne-Mumford stacks. Assume that $X$ is integral, and assume that $X$ has rational singularities: there exists a surjective, étale morphism $g:\widetilde{X}\to X$ such that $\widetilde{X}$ is a $k$-scheme that has rational singularities. Finally, assume that the geometric generic fiber of $f$ is isomorphic (as a stack) to a smooth proper variety that is rationally connected, or even just $\mathcal{O}$-acyclic.

**Proposition**. The natural map $\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism, and $R^q f_*\mathcal{O}_X$ equals $0$ for every $q>0$.

Because $h$ is flat, $h^*R^qf_*\mathcal{O}_X$ equals $R^q \widetilde{f} \mathcal{O}_{X\times_Y \widetilde{Y}}$, where $\widetilde{f}:X\times_Y \widetilde{Y} \to \widetilde{Y}$ is the projection. Thus, without loss of generality, assume that $Y$ is a normal scheme. As in the comment, there exists a coarse moduli space $p:X \to |X|$. The morphism $f$ factors as the composition of $p$ and a surjective, proper $k$-morphism, $$|f|:|X|\to Y.$$ Because the characteristic is $0$, $X$ is a tame stack. Thus, $\mathcal{O}_{|X|} \to p_*\mathcal{O}_X$ is an isomorphism and $R^qp_*\mathcal{O}_X$ equals $0$ for all $q>0$. Therefore, it suffices to prove that $\mathcal{O}_Y\to |f|_*\mathcal{O}_{|X|}$ is an isomorphism and $R^q|f|_*\mathcal{O}_{|X|}$ equals $0$ for all $q>0$. Via Buch-Mihalcea, it suffices to prove that $|X|$ has rational singularities. This follows from Proposition 5.13, p. 157 of "Birational Geometry of Algebraic Varieties" by Kollár and Mori.