Volunteer-based food rescue platforms tackle food waste by matching surplus food to communities in need. These platforms face the dual problem of maintaining volunteer engagement and maximizing the food rescued. Existing algorithms to improve volunteer engagement exacerbate geographical disparities, leaving some communities systematically disadvantaged. We address this issue by proposing Contextual Budget Bandit. Contextual Budget Bandit incorporates context-dependent budget allocation in restless multi-armed bandits, a model of decision-making which allows for stateful arms. By doing so, we can allocate higher budgets to communities with lower match rates, thereby alleviating geographical disparities. To tackle this problem, we develop an empirically fast heuristic algorithm. Because the heuristic algorithm can achieve a poor approximation when active volunteers are scarce, we design the Mitosis algorithm, which is guaranteed to compute the optimal budget allocation. Empirically, we demonstrate that our algorithms outperform baselines on both synthetic and real-world food rescue datasets, and show how our algorithm achieves geographical fairness in food rescue.