The electrolysis of CO₂ converts CO₂ into value-added products and has the potential to reduce the use of fossil feedstocks by serving as a circular alternative in chemical processes. However, existing literature lacks comprehensive and quantitative analyses of economic, environmental, and governmental factors that can hinder or support its deployment. This study addresses this gap by exploring the potential of CO₂ electrolysis for the production of synthesis gas, syngas, through a supply chain design that integrates long-term decisions on location and infrastructure and medium-term decisions on capacity expansion and aggregate production planning. We identify and quantify time-dependent and uncertain parameters using the Delphi method and employ multi-period planning and robust optimization approaches to consider them, respectively. Moreover, since the environmental impact of syngas is highly dependent on electricity consumption, renewable electricity is utilized with battery support alongside grid electricity. Accordingly, we propose a mixed integer linear programming model to design a supply chain that can serve as a benchmark to make CO₂ electrolysis financially and environmentally viable for syngas production. We conduct a case study on the Benelux region, analyzing different scenarios to derive managerial and design insights. The results show that a design that takes uncertainty into account can reduce syngas production costs by up to 22%. Additionally, although renewable electricity supply variability and different grid characteristics across countries can lead to different strategic decisions with higher costs, increased battery installations and higher government financial support for renewable electricity can help eliminate differences in designs.