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Recent studies and industry advancements indicate that modular vehicles (MVs) have the potential to enhance transportation systems through their ability to dock and split during a trip. Although various applications of MVs have been explored across different domains, their application in logistics remains underexplored. This study examines the use of MVs in cargo delivery to reduce total delivery costs. We model the delivery problem for MVs as a variant of the Vehicle Routing Problem, referred to as the Modular Vehicle Routing Problem (MVRP). In the MVRP, MVs can either serve customers independently or dock with other MVs to form a platoon, thereby reducing the average cost per unit. In this study, we mainly focus on two fundamental types of MVRPs, namely the capacitated MVRP (CMVRP) and the MVRP with time windows (MVRPTW). To address these problems, we first developed mixed-integer linear programming (MILP) models, which can be solved using commercial optimization solvers. Given the NP-hardness of this problem, we also designed a Tabu Search (TS) algorithm with a solution representation based on Gantt charts and a neighborhood structure tailored for the MVRP. Multi-start and shaking strategies were incorporated into the TS algorithm to escape local optima. Additionally, we explored other potential applications in logistics and discussed problem settings for three MVRP variants. Results from numerical experiments indicate that the proposed algorithm successfully identifies nearly all optimal solutions found by the MILP model in small-size benchmark instances, while also demonstrating good convergence speed in large-size benchmark instances. Comparative experiments show that the MVRP approach can reduce costs by approximately 5.6% compared to traditional delivery methods. Sensitivity analyses reveal that improving the cost-saving capability of MV platooning can enhance overall benefits.