  Coherent exclusive vector meson electroproduction is a key physics channel at
an electron-ion collider. It probes the gluon structure of nuclei over a wide
range of $Q^2$, and can be used to measure nuclear shadowing and to search for
gluon saturation and/or the colored glass condensate. In this paper, we present
calculations of the kinematic distributions for a variety of exclusive vector
meson final states: the $\rho$, $\phi$, J/$\psi$, $\psi'$ and the $\Upsilon$
states. The cross-sections for light and $c\overline c$ mesons are large, while
$\Upsilon$ states should be produced in moderate numbers at a medium energy EIC
(the proposed U.S. designs) and in large numbers at the LHeC. We investigate
the acceptances for these states, as a function of detector rapidity coverage.
A large-acceptance detector is needed to cover the full range photon-nucleon
collision energies produced at an EIC; a forward detector is required to
observe vector mesons from the most energetic photon interactions, and thereby
probe gluons at the lowest possible Bjorken-$x$ values.
