  In this paper we analyze azimuthal asymmetries in the processes of
unpolarized and polarized $J/\psi \,(\Upsilon)$ production at an Electron-Ion
Collider. Apart from giving access to various unknown gluon transverse momentum
distributions, we suggest to use them as a new method to extract specific
color-octet NRQCD long-distance matrix elements, i.e.\ $\langle0|{\cal
O}_{8}^{J/\psi}(^{1}S_{0})|0\rangle$ and $\langle0|{\cal
O}_{8}^{J/\psi}(^{3}P_{0})|0\rangle$, whose values are still quite uncertain
and for which lattice calculations are unavailable. The new method is based on
combining measurements of analogous asymmetries in open heavy-quark pair
production which can be performed at the same energy. To enhance the gluon
contribution one can consider smaller values of $x$ and, in order to assess the
impact of small-$x$ evolution, we perform a numerical study using the MV model
as a starting input and evolve it with the JIMWLK equations.
