Exclusive vector meson electroproduction over a broad $Q^2$ range offers a unique opportunity to probe the gluon structure of nuclei to measure nuclear shadowing, and to search for gluon saturation and/or the colored glass condensate at an Electron-Ion Collider. Understanding the kinematic distributions and cross sections for specific processes can impact detector design to maximize their acceptance and strengthen the physics case. We will discuss predictions from a Monte Carlo generator eSTARlight, a tool developed to study production of exclusive vector meson final states. We present final state distributions and production rates for the $\rho$, $\phi$, J/$\psi$, $\psi'$ and the $\Upsilon$ states in $ep$ and $eA$ collisions at the different energies.