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This article proposes a distributed control method for networked manipulators to cooperatively transport an unmodeled object without force measurement. First, we design an adaptive neural network control algorithm, based on the wavelet neural network, to estimate unknown system dynamics. A novel task-space synchronization control is studied with triggering condition for the network under time-varying communication delays, to improve the cooperative quality while reducing the network burden. The proposed control method ensures uniform ultimate boundedness of velocity/position tracking errors and boundedness of the internal force with the avoidance of Zeno behavior. Moreover, the robustness of networked uncertain manipulators is guaranteed. Finally, a numerical simulation and experiments are carried to demonstrate the performance and efficiency of the proposed control algorithm.