Motivated by the need to create plans for delivering medication quickly during a public health emergency, this paper formulates the Inventory Slack Routing Problem. The planning goal is to deliver material as early as possible to demand sites from a central depot at which material arrives over time. The objective function is to maximize the minimum slack of the deliveries so that all demand sites are treated equitably. This paper presents and analyzes the problem, discusses solution techniques, and discusses the results of computational experiments used to compare the solution techniques. Although motivated by planning for public health emergencies, this work is also applicable to other settings in which material must be delivered quickly to multiple facilities that rely upon the material to operate.