Although supply chain network design under uncertainty has been studied by many researchers, most stochastic programming approaches in this area assume uncertain parameters follow certain distribution functions. However, in practice, the true distributions may be ambiguous and some historical data are available. This study proposes a data-driven two-stage stochastic programming model to obtain robust decisions among all possible distributions in a defined ambiguity set based on the moments of available data. In accordance with the proposed stochastic program, a solution algorithm based on Benders’ decomposition is developed. Further, the social concerns corresponding to the supply chain network are derived and quantified by the social life cycle assessment methodology. The proposed model is applied for designing a recovery network in which various technologies use generated municipal solid wastes for the power generation. Computational results on a real-life case study demonstrate the applicability of the proposed data-driven two-stage stochastic model as well as the impact of considering social concerns on the design decisions.