Vanishing of higher direct image of finite morphisms relative to the fppf topology

No, it is not true. Let $k$ be an algebraically closed field of characteristic $p > 0$ and set $k' := k[x]/(x^2)$. Let $f \colon \mathrm{Spec}(k') \rightarrow \mathrm{Spec}(k)$ be the corresponding map. Then the sequence $0 \rightarrow \mu_p \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$ will show you that $(R^1f_*(\mu_p))(k) \cong k'^{\times}/k'^{\times p} \neq 0$.

As far as I know, it is an open question whether such vanishing is true when $f$ is a closed immersion.