  We calculate the matching, at leading order, of the transverse
momentum-dependent fragmentation functions (TMDFFs) for light quarks and gluons
fragmenting to a $J/\psi$ onto polarized nonrelativistic QCD (NRQCD) TMDFFs.
The NRQCD TMDFFs have an operator-product-expansion in terms of nonperturbative
NRQCD production matrix elements. Using the results we obtain, we make
predictions for the light quark fragmentation contribution to the production of
polarized $J/\psi$ in semi-inclusive deep inelastic scattering (SIDIS) both for
unpolarized and longitudinally polarized beams. These results are an important
contribution to polarized $J/\psi$ production in SIDIS, and thus are needed for
comparison with experiments at the future Electron-Ion Collider.
