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Nonprofit organizations (NPOs) lack resources, hindering the quality and quantity of service they can deliver. Meanwhile, NPOs at times have underutilized or even spare resources due to the inability to scale expertise in staffing and tangible resources to meet temporally shifting service demands. These observations motivate us to propose a novel resource sharing system, SWAP, which to the best of our knowledge, is the first resource sharing system that facilitates resource exchanges where NPOs can obtain resources by offering their own. SWAP consists of four elements: a collaborative auction-based sharing process, complete with an offering mechanism, a bidding mechanism, and the virtual currency, SWAPcredit, to facilitate liquidity in exchange; a central technology that represents the award determination problem with a multilateral exchange optimization model, generating resource exchange outcomes; an online platform, the SWAP Hub, where NPOs can offer and bid on available resources, and receive exchange results; and human-centric co-design, shaping the understanding and design decisions of a research collective, that includes the authors and NPO professionals. We conduct a series of experiments using both empirical and simulated data to illustrate the benefits and potential of SWAP. Our results demonstrate that SWAP can address temporal resource needs in practice; show that optimal exchange outcomes can be generated even for large-scale SWAP markets; and provide strong evidence in support of guidance to inform the progression for future versions of SWAP. The SWAP system is presently implemented in Howard County, MD, USA, with ongoing enhancements and potential for future expansion.