Mobility services operate in environments characterized by time-varying and spatially heterogeneous demand. Modular vehicle systems enable flexible operations by allowing independent units to couple and decouple, forming platoons that can change size over time. This flexibility has the potential to improve adaptability to varying demand and enhance overall service efficiency in demand-responsive transportation settings. The problem is formulated as a Mixed-Integer Linear Programming model that jointly determines vehicle routing, module attach and detach operations, and passenger service under time window constraints, aiming to minimize total operational costs while maintaining a high level of service. To improve computational efficiency while preserving solution quality, a two-phase heuristic is proposed, which first determines an initial routing and request-assignment solution and subsequently exploits full modularity through a Large Neighborhood Search procedure.