We consider the functional regular variation in the space $\mathbb{D}$ of c\`adl\`ag functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int\int f(A, t - s) \Lambda (d A, d s)$. We give sufficient conditions for an MMA process $(X_t)$ to have c\`adl\`ag sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb{D}$ if the driving L\'evy basis is regularly varying and the kernel function $f$ satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered in detail.