This paper investigates a communication-constrained unmanned aerial vehicle (UAV) pickup and delivery system for continuous multi-period operations. To ensure real-time control updates between UAVs and the ground server, a minimum communication rate requirement is imposed throughout each mission. The objective is to minimize the average mission completion time of multiple rotary-wing UAVs while satisfying mobility, payload, safety, and communication constraints. The resulting mixed-integer nonlinear programming problem, involving binary pickup/drop-off decisions, trajectories, and variable time-slot durations, is mathematically intractable. To address this, a successive convex approximation framework combined with a penalty convex–concave procedure is developed, enabling iterative convex reformulation and convergence to a near-optimal binary-feasible solution. Simulation results demonstrate that the proposed algorithm efficiently generates collision-free trajectories and adaptive flight paths that maintain reliable communication links, outperforming baseline strategies in terms of completion time and coordination efficiency under communication constraints.