We study the exclusive production of $J^{PC}=0^{++}, 0^{--}$ charmonium states in proton-proton collisions at the LHC energies The $pp to ppeta_c$ reaction is discussed for the first time. We observe a substantial contribution from the nonperturbative domain of gluon virtualities, especially for $eta_c$ production. To model the nonperturbative region better, we utilize models of the unintegrated gluon distribution based on parametrizations of the color dipole cross-section.