Carbon monoxide (CO) emission has been observed in a number of core-collapse supernovae (SNe) and is known to be an important coolant at late times. We have implemented a chemical reaction network in the radiative-transfer code CMFGEN to investigate the formation of CO and its impact on SN ejecta. We calculate two 1D SN models with and without CO: a BSG explosion model at one nebular epoch and a full time sequence (50 to 300 days) for a RSG explosion. In both models, CO forms at nebular times in the dense, inner regions at velocities $<2000 \mathrm{km/s}$ where line emission from CO can dominate the cooling and reduce the local temperature by as much as a factor of two, weakening emission lines and causing the optical light curve to fade faster. That energy is instead emitted in CO bands, primarily the fundamental band at $\sim 4.5\mathrm{\mu m}$, which accounts for up to 20% of the total luminosity at late times. However, the non-monotonic nature of the CO cooling function can cause numerical difficulties and introduce multiple temperature solutions. This issue is compounded by the sensitivity of the CO abundance to a few reaction rates, many of which have large uncertainties or disparate values across literature sources. Our results also suggest that, in many SNe, CO level populations are far from their LTE values. Unfortunately, accurate collisional data, necessary to compute NLTE populations, are limited to a few transitions.