Medical image segmentation in X-ray images is beneficial for computer-aided diagnosis and lesion localization. Existing methods mainly fall into a centralized learning paradigm, which is inapplicable in the practical medical scenario that only has access to distributed data islands. Federated Learning has the potential to offer a distributed solution but struggles with heavy training instability due to client-wise domain heterogeneity (including distribution diversity and class imbalance). In this paper, we propose a novel Federated Client-tailored Adapter (FCA) framework for medical image segmentation, which achieves stable and client-tailored adaptive segmentation without sharing sensitive local data. Specifically, the federated adapter stirs universal knowledge in off-the-shelf medical foundation models to stabilize the federated training process. In addition, we develop two client-tailored federated updating strategies that adaptively decompose the adapter into common and individual components, then globally and independently update the parameter groups associated with common client-invariant and individual client-specific units, respectively. They further stabilize the heterogeneous federated learning process and realize optimal client-tailored instead of sub-optimal global-compromised segmentation models. Extensive experiments on three large-scale datasets demonstrate the effectiveness and superiority of the proposed FCA framework for federated medical segmentation.