Locate-then-Edit Knowledge Editing (LEKE) is a key technique for updating large language models (LLMs) without full retraining. However, existing methods assume a single-user setting and become inefficient in real-world multi-client scenarios, where decentralized organizations (e.g., hospitals, financial institutions) independently update overlapping knowledge, leading to redundant mediator knowledge vector (MKV) computations and privacy concerns. To address these challenges, we introduce Federated Locate-then-Edit Knowledge Editing (FLEKE), a novel task that enables multiple clients to collaboratively perform LEKE while preserving privacy and reducing computational overhead. To achieve this, we propose FedEdit, a two-stage framework that optimizes MKV selection and reuse. In the first stage, clients locally apply LEKE and upload the computed MKVs. In the second stage, rather than relying solely on server-based MKV sharing, FLEKE allows clients retrieve relevant MKVs based on cosine similarity, enabling knowledge re-edit and minimizing redundant computations. Experimental results on two benchmark datasets demonstrate that FedEdit retains over 96% of the performance of non-federated LEKE while significantly outperforming a FedAvg-based baseline by approximately twofold. Besides, we find that MEMIT performs more consistently than PMET in the FLEKE task with our FedEdit framework. Our code is available at https://github.com/zongkaiz/FLEKE.