Abstract: Federated zeroth-order optimization (FedZOO) reduces the memory and communication costs of collaborative training for edge intelligence by avoiding backpropagation while communicating only random seeds and scalar coefficients. However, existing methods lack efficient mechanisms for exact unlearning to fully remove the influence of training data when users request deletion. A key insight is that total variation stability of the learning algorithm creates overlap between model distributions before and after data deletion, allowing exact unlearning to reuse computation through coupling. Based on this observation, we jointly design two stable zeroth-order learning algorithms and their corresponding exact-unlearning procedures: \textsc{FedZOO-S} obtains stability through client and mini-batch subsampling, enabling \textsc{FedZOUL-S} to perform efficient unlearning by preserving earlier updates and restarting from the first affected step. \textsc{FedZOO-NS} further combines subsampling with Gaussian noise to increase overlap between update distributions; \textsc{FedZOUL-NS} exploits this overlap through rejection--reflection coupling to reuse affected updates when the coupling is accepted. We prove exact unlearning for both algorithms: the model and saved state have the same joint distribution as retraining with the corresponding learner on retained data. We further establish stability and convergence guarantees for the learning algorithms and derive upper bounds on expected unlearning cost. Experiments demonstrate effective unlearning with competitive model utility and up to $56.1\%$ less cumulative unlearning time than re-optimization. Compared with first-order baselines, our methods reduce communication overhead by up to approximately four orders of magnitude.