  We compute exclusive $\eta_c$ production in high energy electron-nucleon and
electron-nucleus collisions that is sensitive to the Odderon. In perturbative
QCD the Odderon is a $C$-odd color singlet consisting of at least three
$t$-channel gluons exchanged with the target. By using the Color Glass
Condensate effective theory our result describes the Odderon exchange at the
high collision energies that would be reached at a future electron-ion
collider. The Odderon distribution is evolved to small-$x$ using the
Balitsky-Kovchegov evolution equation with running coupling corrections. We
find that while at low momentum transfers $t$ the cross section off a proton is
dominated by the Primakoff process, the Odderon becomes relevant at larger
momentum transfers of $|t|\geq1.5$ GeV$^2$. We point that the Odderon could
also be extracted at low-$t$ using neutron targets since the Primakoff
component is strongly suppressed. In the case of nuclear targets, the Odderon
cross section becomes enhanced thanks to the mass number of the nuclear target.
The gluon saturation effect induces a shift in the diffractive pattern with
respect to the Primakoff process that could be used as a signal for the
Odderon.
