Distributed edge computing systems that participate in emergency demand response (EDR) programs can adjust workload across heterogenous edges to reduce total energy consumption. Unfortunately, this approach may not always reduce sufficient energy as required by EDR. In this article, we propose to leverage Electrical Vehicles (EVs) and Vehicle-to-Grid (V2G) techniques to provide energy to the edge system, and design an auction mechanism to incentivize EVs to discharge energy for the edges. Yet, we face critical challenges, such as the uncertainty of EV bid arrivals, the restriction of discharging deadlines, and the desire to achieve required economic efficiency. To overcome such challenges, we design a novel online approach, $E^{3}$DR, of multiple algorithms that decompose our original NP-hard social cost minimization problem into two subproblems, solve the first subproblem via reformulation, the primal-dual optimization theory, and a careful payment design, and solve the second subproblem via standard solvers. We have rigorously proved that our approach finishes in polynomial time, achieves truthfulness and individual rationality economically, and leads to a parameterized competitive ratio for the long-term social cost. Through extensive evaluations using real-world data traces, we have validated the superior practical performance of our approach compared to existing algorithms.