This paper presents a distributed control approach for time-varying formation of heterogeneous planar underactuated vehicle networks without global position measurements. All vehicles in the network are modeled as generic three degree-of-freedom planar rigid bodies with two control inputs, and are allowed to have nonidentical dynamics. Feasible trajectories are generated for each vehicle using the nonholonomic constraints of the vehicle dynamics. By exploiting the cascaded structure of the planar vehicle model, a transformation is introduced to define the reduced order error dynamics, and then, a sliding-mode control law is proposed. Low-level controller for each vehicle is derived such that it only requires relative position and local motion information of its neighbors in a given directed communication network. The proposed formation control law guarantees the uniform global asymptotic stability (UGAS) of the closed-loop system subject to bounded uncertainties and disturbances. The proposed approach can be applied to underactuated vehicle networks consisting of mobile robots, surface vessels, and planar aircraft. Simulations are presented to demonstrate the effectiveness of the proposed control scheme.