Central to the contemporary American welfare state, nonprofits are increasingly squeezed between declining resources and a simultaneous peak in demand for services. Scholars have outlined a number of ways that nonprofits strategically respond to resource scarcity. Why nonprofits select particular strategies, however, is less clear. This article motivates a network membership perspective via a mixed-methods study of one food bank and its associated food pantries, wherein actors navigate competing network demands that are emblematic of those faced by other nonprofits. Food banks source food from three types of networks - a cartel-like network orchestrated by a national coordinating body, a peer-to-peer inter-food bank network, and a bureaucratic network anchored by the USDA - that place overlapping and conflicting constraints on the organizations. In combination, network demands entangle the food banks within a web of tensions that prevent many forms of adaptation while enabling others, a predicament I refer to in this article as network entrapment. How organizations navigate network demands has implications for the vitality of the organizations and the wellbeing of their beneficiaries. My results suggest that entrapment can lead to organizational burden and the adoption of neoliberal-inspired efficiencies, but also innovations that prioritize citizen rights over cost-benefit ratios.