While vaccinations continue to be rolled out to curb the ongoing COVID-19 pandemic, their verification is becoming a requirement for the re-incorporation of individuals into many social activities or travel. Blockchain technology has been widely proposed to manage vaccination records and their verification in many politically-bound regions. However, the high contagiousness of COVID-19 calls for a global vaccination campaign. Therefore, a blockchain for vaccination management must scale up to support such a campaign and be adaptable to the requirements of different countries. While there have been many proposals of blockchain frameworks that balance the access and immutability of vaccination records, their scalability, a critical feature, has not yet been addressed. In this paper, we propose a scalable and cooperative Global Immunization Information Blockchain-based System (GEOS) that leverages the global interoperability of immunization information systems. We model GEOS and describe its requirements, features, and operation. We analyze the communications and the delays incurred by the national and international consensus processes and blockchain interoperability in GEOS. Such communications are pivotal in enabling global-scale interoperability and access to electronic vaccination records for verification. We show that GEOS ably keeps up with the global vaccination rates of COVID-19 as an example of its scalability.