Ensuring the integrity of business processes without disclosing confidential business information is a major challenge in inter-organizational processes. This paper introduces a zero-knowledge proof (ZKP)-based approach for the verifiable execution of business processes while preserving confidentiality. We integrate ZK virtual machines (zkVMs) into business process management engines through a comprehensive system architecture and a prototypical implementation. Our approach supports chained verifiable computations through proof compositions. On the example of product carbon footprinting, we model sequential footprinting activities and demonstrate how organizations can prove and verify the integrity of verifiable processes without exposing sensitive information. We assess different ZKP proving variants within process models for their efficiency in proving and verifying, and discuss the practical integration of ZKPs throughout the Business Process Management (BPM) lifecycle. Our experiment-driven evaluation demonstrates the automation of process verification under given confidentiality constraints.