Food waste and insecurity are two societal challenges that coexist in many parts of the world. A prominent force to combat these issues, food rescue platforms match food donations to organizations that serve underprivileged communities, and then rely on external volunteers to transport the food. Previous work has developed machine learning models for food rescue volunteer engagement. However, having long worked with domain practitioners to deploy AI tools to help with food rescues, we understand that there are four main pain points that keep such a machine learning model from being actually useful in practice: small data, data collected only under the default intervention, unmodeled objectives due to communication gap, and unforeseen consequences of the intervention. In this paper, we introduce bandit data-driven optimization which not only helps address these pain points in food rescue, but also is applicable to other nonprofit domains that share similar challenges. Bandit data-driven optimization combines the advantages of online bandit learning and offline predictive analytics in an integrated framework. We propose PROOF, a novel algorithm for this framework and formally prove that it has no-regret. We show that PROOF performs better than existing baseline on food rescue volunteer recommendation.