In a context where companies are striving to produce highly customised goods in small batches and within short lead times, increasing attention is being put on the design and management of part feeding systems. This research is the first to model automated part feeding to supermarkets in a factory environment, considering an innovative technology called vertical robotic storage and retrieval systems. This technology allows automating the storage, picking, and internal transportation activities in an integrated process, thanks to rack-climbing robots roaming in both the shop floor and the storage racks. We develop an analytical model based on the queuing network approach to analyse the system performance, and we use it to perform numerical experiments and to evaluate the design trade-offs with reference to a real case in the automotive industry. Results show that an increase in the number of robots leads to better performance since the positive impact on the response time is stronger than the negative impact on the waiting times of robots at the supermarkets due to congestion. Furthermore, a configuration with multiple small supermarkets improves the efficiency of the replenishment process, compared to a setting with few big supermarkets.