Large language models (LLMs) have demonstrated strong performance on medical benchmarks, including question answering and diagnosis. To enable their use in clinical settings, LLMs are typically further adapted through continued pretraining or post-training using clinical data. However, most medical LLMs are trained on data from a single institution, which faces limitations in generalizability and safety in heterogeneous systems. Federated learning (FL) is a promising solution for enabling collaborative model development across healthcare institutions. Yet applying FL to LLMs in medicine remains fundamentally limited. First, conventional FL requires transmitting the full model during each communication round, which becomes impractical for multi-billion-parameter LLMs given the limited computational resources. Second, many FL algorithms implicitly assume data homogeneity, whereas real-world clinical data are highly heterogeneous across patients, diseases, and institutional practices. We introduce the model-agnostic and parameter-efficient federated learning framework for adapting LLMs to medical applications. Fed-MedLoRA transmits only low-rank adapter parameters, reducing communication and computation overhead, while Fed-MedLoRA+ further incorporates adaptive, data-aware aggregation to improve convergence under cross-site heterogeneity. We apply the framework to clinical information extraction (IE), which transforms patient narratives into structured medical entities and relations. Accuracy was assessed across five patient cohorts through comparisons with BERT models, and LLaMA-3 and DeepSeek-R1, GPT-4o models. Evaluation settings included (1) in-domain training and testing, (2) external validation on independent cohorts, and (3) a low-resource new-site adaptation scenario using real-world clinical notes from the Yale New Haven Health System.