Large-scale electric vehicle (EV) adoption has the potential to offer flexible and distributed energy resources to support the power distribution system (DS) during emergent incidents. We have developed a network-based multi-agent stochastic optimization model with equilibrium constraints (N-MSOPEC) to investigate the potential value of EVs on the emergency load pickup in DS, considering the uncertainties of line outage and EV participation. Decentralized stakeholders from both transportation and DSs, including distribution system operator (DSO), distributed generator (DG) owners, charging station aggregator (CSA), and EV drivers, are explicitly modeled to reflect realistic decentralized decision-making. EV participation is incentivized in a market equilibrium framework for DS support. Additionally, an exact convex reformulation technique has been developed for the proposed N-MSOPEC model, which significantly improves the computational efficiency to solve high-dimensional complementarity problems. Simulation results on coupled transportation and distribution test systems showed how EV participation could reduce load loss in different line outages and EV adoption scenarios and demonstrated the effectiveness of our model in capturing the interdependencies of the coupled systems. The system characteristics and load pickup needs of DS influenced the incentives provided for EV drivers, resulting in the different CS selections and load pickup patterns. Additionally, we investigated the value of stochastic modeling in the multi-agent framework by calculating stochastic metrics such as the value of stochastic solution (VSS) and expected value of perfect information (EVPI) for the involved stakeholders.