By flat base change we may assume that $k$ is algebraically closed. All fibers of $f:\mathcal{X}\to S$ are Cohen-Macaulay since $\mathcal{X}$ is regular, and the desired result is equivalent to $R^{q}f_{*}(\omega_{\mathcal{X}}\otimes\mathcal{L})=0$ for $q\geq 1$ since $S$ is affine. We may therefore use relative duality for $f$, which shows that the desired result is equivalent to $R^{q}f_{*}\mathcal{L}^{-1}=0$ for $q<n=\dim X$.

Let $d\in\mathbf{N}^{*}$ be a common multiple of the multiplicities of $\mathcal{X}_{0}$, set $S_{d}:=\spec k[[t^{1/d}]]$ and let $\mathcal{Y}$ be the normalization of $\mathcal{X}\times_{S}S_{d}$, with structure map $g:\mathcal{Y}\to S$. The pull-back $\mathcal{M}$ of $\mathcal{L}$ to $\mathcal{Y}$ is still ample since $\mathcal{Y}\to\mathcal{X}$ is finite. By [KKMS, pp.200–201] the $S$-scheme $\mathcal{Y}$ is toroidal and its special fiber $\mathcal{Y}_{0}$ is reduced.

The relative trace $\tr_{\mathcal{Y}/\mathcal{X}}$ shows that $R^{q}g_{*}\mathcal{M}^{-1}$ contains $R^{q}f_{*}\mathcal{L}^{-1}$ as a direct summand, and it is therefore enough to show by semicontinuity that $H^{q}(\mathcal{Y}_{0},\mathcal{M}^{-1})=0$ for $q<n$. Like any toroidal $S$-scheme, $\mathcal{Y}$ is Cohen-Macaulay. As a consequence, the Cartier divisor $\mathcal{Y}_{0}$ is Cohen-Macaulay as well. By another application of duality, this time on $\mathcal{Y}_{0}$, we are reduced to showing that $H^{q}(\mathcal{Y}_{0},\omega_{\mathcal{Y}_{0}}\otimes\mathcal{M})=0$ for $q\geq 1$.

By [KKMS] we may choose a toroidal vertical blow-up $\pi:\mathcal{Y}^{\prime}\to\mathcal{Y}$ such that $\mathcal{Y}_{0}$ has simple normal crossing support. A toric computation (compare [Kol97, Proposition 3.7]) shows that

Since $\mathcal{Y}_{0}$ and $\mathcal{Y}^{\prime}_{0,\mathrm{red}}$ are Cartier divisors on $\mathcal{Y}$ and $\mathcal{Y}^{\prime}$ respectively, adjunction applies (see for instance [KM98, Proposition 5.73]) and we get $\omega_{\mathcal{Y}^{\prime}_{0,\mathrm{red}}}\simeq\pi^{*}\omega_{\mathcal{Y}_{0}}$. On the other hand the projective reduced (but a priori reducible) $k$-scheme $\mathcal{Y}^{\prime}_{0,\mathrm{red}}$ has embedded SNC singularities. It is indeed an SNC divisor in $\mathcal{Y}^{\prime}$, and [Art69] implies that the existence of an algebraic $k$-variety containing $\mathcal{Y}^{\prime}_{0}$ as a divisor. Since $\pi:\mathcal{Y}^{\prime}_{0,\mathrm{red}}\to\mathcal{Y}_{0}$ is projective and $\mathcal{M}$ is ample on $\mathcal{Y}_{0}$, we may therefore apply a vanishing theorem originally due to Kawamata and Ambro and corrected by Fujino ([Kaw85, Theorem 4.4], [Amb03, Theorem 3.2] and [Fuj09, Theorem 2.39]) to get that

is acyclic on $\mathcal{Y}_{0}$. ∎