We initiate the study of fair distribution of delivery tasks among a set of agents wherein delivery jobs are placed along the vertices of a graph. Our goal is to fairly distribute delivery costs (modeled as a submodular function) among a fixed set of agents while satisfying some desirable notions of economic efficiency. We adopt well-established fairness concepts—such as envy-freeness up to one item (EF1) and minimax share (MMS)—to our setting and show that fairness is often incompatible with the efficiency notion of social optimality. Yet, we characterize instances that admit fair and socially optimal solutions by exploiting graph structures. We further show that achieving fairness along with Pareto optimality is computationally intractable. Nonetheless, we design an XP algorithm (parameterized by the number of agents) for finding MMS and Pareto optimal solutions on every tree instance, and show that the same algorithm can be modified to find efficient solutions along with EF1, when such solutions exist. We complement these results by theoretically and experimentally analyzing the price of fairness.