The computing frontier is moving from centralized mega datacenters towards distributed cloudlets at the network edge. We argue that cloudlets are well-suited for participation in Emergency Demand Response (EDR) programs due to their enormous energy consumption and flexible workload distribution, while existing EDR mechanisms for clouds and colocation datacenters are not suitable for cloudlets. We propose a novel online market mechanism, EdgeEDR, to incentivize cloudlets to participate in EDR, featuring multiple cloudlet-specific designs. At a high level, we observe that cloudlet operators can dynamically switch on/off entire cloudlets to compensate for the energy reduction required by the power grid. We formulate a long-term social cost minimization problem and decompose it into a series of one-round procurement auctions. In each auction instance, we propose to let the cloudlet tenants bid with cost functions of their service quality degradation tolerance, and let the cloudlet operator choose the service quality, allocate the workload, and shut down the cloudlets. Via rigorous analysis, we exhibit that our bidding policy is individually rational and truthful; our workload distribution algorithm has near-optimal performance in each auction; and our overall online algorithm achieves a provable competitive ratio. We further confirm the performance of our mechanism through extensive trace-driven simulations.