In this study, a novel sequential broad learning (SBL) approach is proposed to reconstruct the missing signal of damaged sensors in structural health monitoring (SHM) sensory networks. It is capable to reconstruct the structural response and external excitation of linear/nonlinear time-varying dynamical systems under stationary/nonstationary excitation for sensory networks. The proposed approach is a model-free data-driven machine learning methodology and the data reconstruction is executed sequentially with moving time windows. The learning algorithm is developed by adopting the recently developed broad learning system (BLS) (Chen and Liu, 2018). In contrast to deep learning that suffers from excessive computational cost for training the stacks of hierarchical layers, BLS is established with a broadly expandable network and can be modified incrementally based on the inherited results from the previous trained architecture. Therefore, BLS provides a computationally very efficient alternative to deep learning. Taking the benefit of BLS, the proposed SBL approach can efficiently handle the massive data stream generated in long-term monitoring. To demonstrate the efficacy and applicability of the proposed approach, simulated examples that cover linear and nonlinear time-varying dynamical systems subjected to stationary/nonstationary wind-load/base excitation with different types of sensing devices are discussed. Moreover, the SHM database of the field measurement monitored from the MIT Green Building is utilized to examine the performance of the proposed approach in realistic application. It is demonstrated that the proposed approach offers a powerful data reconstruction tool for challenging data missing situations encountered in SHM.