This paper explores the logistics operations of instant grocery delivery services. Therefore, we introduce the instant delivery problem (IDP) to replicate and examine two widely adopted strategies in the rapid delivery market: the personal shopper system (PSS) and the inventory owned delivery (IOD) system. In the PSS, couriers visit affiliated brick- and-mortar stores in the delivery area to pick up and purchase ordered products and then deliver them to customers. In the IOD system, couriers collect products from a single distribution center, or so-called dark store, in which the platform manages the inventory. Even though a PSS strategy is asset light because of the utilization of existing retailers in the area, maintaining a good level of on-time instant deliveries with the PSS is more complex than with IOD. This is because the PSS requires deciding which store to purchase ordered goods from, and picking and shopping at stores needs to be considered in the real-time decision process. We propose a tailored rolling horizon framework that utilizes column generation to browse updated delivery plans for arriving customer orders. Computational studies both in real life–inspired settings and in case studies on selected urban areas show that the PSS is a highly competitive strategy compared with IOD, particularly when dealing with small-sized customer orders. We observe that the performance of the PSS is robust when varying the delivery service time frame. The case studies also suggest that the PSS becomes even more competitive in areas where the retail store density is high.