CO2 utilization to produce value-added products such as cyclic carbonates, especially under mild catalytic conditions, is a promising strategy to reduce CO2 emissions to the environment. In this work, nitrogen-rich, click-based porous polymers (CPPs) were prepared from coupling of tripropargylamine and rigid diazido aromatic substrates, featuring 1,4-phenylene (TB), 1,5-naphthalene (TN), and 4,4′-benzanilide (TBA) linkers, in moderate to good yields (48–83%) using Cu-catalyzed azide–alkyne cycloaddition (CuAAC; TB-C, TN-C, TBA-C) and Huisgen cycloaddition (TB-H, TN-H, and TBA-H). The presence of the chelate tris(triazolylmethyl)amine ligands within the polymer's structure led to strong copper binding and consequently high copper loading ability of the CPPs. In addition, high contents of Lewis basic N donors (21–27 wt% N) of TB-C, TN-C, TBA-C are most likely attributed to strong interactions between CO2 and CPPs and exceptionally high CO2 selectivity over N2 at 273 K, based on gas sorption studies. The catalyst system TB-C/TBAB (TBAB = tetrabutylammonium bromide) promoted complete conversion of epoxides to cyclic carbonates under an atmospheric CO2 pressure, after 24 h or 48 h at 100 °C, with the exception of the epichlorohydrin substrate where 42% selectivity was observed by 1H NMR spectroscopy. The TB-C/TBAB catalyst system could be reused for five catalytic runs without a loss in catalytic activities and any pre-activation processes.