Creating robots that can perform general-purpose service tasks in a human-populated environment has been a longstanding grand challenge for AI and Robotics research. One particularly valuable skill that is relevant to a wide variety of tasks is the ability to locate and retrieve objects upon request. This paper models this skill as a Scavenger Hunt (SH) game, which we formulate as a variation of the NP-hard stochastic traveling purchaser problem. In this problem, the goal is to find a set of objects as quickly as possible, given probability distributions of where they may be found. We investigate the performance of several solution algorithms for the SH problem, both in simulation and on a real mobile robot. We use Reinforcement Learning (RL) to train an agent to plan a minimal cost path, and show that the RL agent can outperform a range of heuristic algorithms, achieving near optimal performance. In order to stimulate research on this problem, we introduce a publicly available software stack and associated website that enable users to upload scavenger hunts which robots can download, perform, and learn from to continually improve their performance on future hunts.