The Large (LMC) and Small (SMC) Magellanic Clouds are irregular satellite galaxies of the Milky Way. Both are metal- and dust-poor, although the SMC is significantly poorer in both. We have recently simulated the chemistry in cold dense regions of the LMC and found that a rich chemistry exists in the gas-phase. In this paper, we report a companion study of the chemistry of dense regions of the SMC, confining our attention to cold regions of dense clouds with a variety of densities, visual extinctions, and grain temperatures, and a fixed gas-phase temperature. With a gas-to-dust ratio and elemental abundances based on observations and scaling, we found that for molecules like CO and N_2_, which are predominantly formed in the gas phase, their abundances are consistent with the reduced elemental abundances of their constituent elements above 25K; however, for species that are produced fully (e.g., CH_3_OH) or partially on the grain surface (e.g., H_2_CO, NH_3_), the dependence on metallicity can be complex. Most of the major gas-phase species observed in our Galaxy are produced in the SMC although in lower quantities. With our simulations, we are able to explain observed gas-phase abundances reasonably well in the dense sources N27 and LIRS 36. We have also compared our calculated abundances of selected ices with limited observations in dense regions in front of young stellar objects.