The evolution of customer requirements has led companies to pursue mass customization and personalization strategies, increasing the complexity of part feeding operations in assembly systems. At the same time, emerging technologies, based on mobile robots, promise to support material handling activities in an efficient yet flexible manner. This paper studies part feeding to assembly stations with Vertical Robotic Storage and Retrieval Systems (VRSRS), a new technology that allows automating both the storage and the internal transportation activities with the same fleet of mobile robots. We develop a Discrete Event Simulation model of a system made of a Central Warehouse replenishing supermarkets which, in turn, supply mixed-model assembly stations. Through this model, we carry out experiments showing a clear trade-off between the number of robots, which affects the overall investment in the system, and the supermarkets size, which affects space occupation in the shop floor, where space is a critical resource. We also estimate the replenishment lead time and the downtime of assembly stations, showing that this system performs best when small production orders are issued and could therefore be a suitable solution to support part feeding activities in a mass customization context.