Cross-hospital collaboration has the potential to address disparities in medical resources across different regions. However, strict privacy regulations prohibit the direct sharing of sensitive patient information between hospitals. Vertical federated learning (VFL) offers a novel privacy-preserving machine learning paradigm that maximizes data utility across multiple hospitals. Traditional VFL methods, however, primarily benefit patients with overlapping data, leaving vulnerable non-overlapping patients without guaranteed improvements in medical prediction services. While some knowledge transfer techniques can enhance the prediction performance for non-overlapping patients, they fall short in addressing scenarios where overlapping and non-overlapping patients belong to different domains, resulting in challenges such as feature heterogeneity and label heterogeneity. To address these issues, we propose a novel unified vertical federated knowledge transfer framework (Unitrans). Our framework consists of three key steps. First, we extract the federated representation of overlapping patients by employing an effective vertical federated representation learning method to model multi-party joint features online. Next, each hospital learns a local knowledge transfer module offline, enabling the transfer of knowledge from the federated representation of overlapping patients to the enriched representation of local non-overlapping patients in a domain-adaptive manner. Finally, hospitals utilize these enriched local representations to enhance performance across various downstream medical prediction tasks. Experiments on real-world medical datasets validate the framework's dual effectiveness in both intra-domain and cross-domain knowledge transfer. The code of \method is available at \url{https://github.com/Chung-ju/Unitrans}.