Resource allocation models in contingency planning aim to mitigate unexpected failures in supply chains due to disruptions with rare occurrence but disastrous consequences. This paper formulates this problems as a two-stage stochastic optimization with a risk-averse recourse function, and proposes a novel computationally tractable solution approach. The method relies on an inexact bundle method and subgradient approximations through a scenario reduction mechanism. It requires solving the second-stage problem only for a small subset of scenarios. We prove that our scenario reduction and function approximations satisfy the requirements of the oracle in the inexact bundle method, ensuring convergence to an optimal solution. The practical performance of the developed inexact bundle method under risk aversion is investigated in the context of a resource allocation problem in contingency planning. Structures of risk-averse optimal solutions for different risk measures and their corresponding optimal values are studied. We then create a library of test problems using real-world data and apply the exact bundle method to find their optimal values. These benchmarks are used to compare the developed solutions under different risk measures and confidence levels. Our analysis indicates that our inexact bundle method significantly reduces the computational time in comparison to the exact bundle method, and is capable to achieve a high percentage of optimality within a much shorter time.