This work locates urgent relief distribution centers (URDCs) on a given set of candidate sites using a robust vertex p-center (RVPC) model. This model addresses uncertain travel times, represented using fixed intervals or ranges instead of probability distributions, between URDCs and affected areas. The objective of locating a predetermined number (p) of URDCs is to minimize worst-case deviation in maximum travel time from the optimal solution. To reduce the complexity of solving the RVPC problem, this work proposes a property that facilitates identification of the worst-case scenario for a given set of URDC locations. Since the problem is NP-hard, a heuristic framework is developed to efficiently obtain robust solutions. Then, a specific implementation of the framework, based on simulated annealing, is developed to conduct computational experiments. Experimental results show that the proposed heuristic is effective and efficient in obtaining robust solutions of interest. This work examines the impact of the degree of data uncertainty on the selected performance measures and the tradeoff between solution quality and robustness. Additionally, this work demonstrates the applicability of the proposed model to natural disasters based on a real-world instance. The result is compared with that obtained by a scenario-based, two-stage stochastic model. This work contributes significantly to the growing body of literature applying robust optimization approaches to emergency logistics.