Since December 2020, the vaccines from several manufacturers, e.g., Pfizer/BioNTech, Moderna, etc., have been approved for mass vaccination to control the COVID-19 pandemic, which has caused more than 100 million infections and 2.4 million deaths. These vaccines are produced and transported in large quantities to suffice the needs of several countries. Before arriving at the end-users, the vaccines need to be stored at extremely low temperatures and distributed through reliable cold chain logistics networks. Thus, the timely and cost-effective distribution of COVID-19 vaccines via cold chain logistics has become a complex operational challenge. In this paper, we develop a simulation-based approach combining both route optimization and dynamic simulation to improve the logistics performance for COVID-19 vaccine distribution. A state-of-the-art simulation package called anyLogistix is used to perform a real-world case study in Norway. With the data of periodic vaccine demands, customer and warehouse locations, vehicle-related costs and emissions, and expected service levels, implications are obtained based on the analysis of several scenarios. Our experimental results reveal that the service level, cost-effectiveness, environmental performance, and equity of a cold chain vaccine logistics system can be significantly influenced by the fleet size, the fleet composition, the type of vehicle used, and the route optimization. Thus, these factors need to be holistically considered in the planning of an effective COVID-19 vaccine distribution system.