Even though Bayesian inference has been widely applied in the field of structural health monitoring (SHM), it often encounters noteworthy challenges if the likelihood function lacks a closed-form expression or is numerically intractable. This is especially true when the computational simulation models involve hierarchically connected sub-models. While likelihood-free approaches have been developed using neural networks to deal with this issue, currently available methods are not suitable for estimating variables that dynamically change with time. This study proposes an innovative likelihood-free inference method for efficient dynamic state estimation using continuously collected data. The proposed framework leverages two complementary neural networks, namely a posterior network and a likelihood network, within the Bayesian framework. The posterior network approximates the posterior distributions for any given observation, while the likelihood network emulates the likelihood of the underlying probabilistic model. These two networks are jointly implemented using conditional invertible neural networks (cINN) based on the normalizing flow. To facilitate continuous model updating over an extended monitoring period, we extended a recursive model updating strategy that was proposed in our previous research to the cINN-based posterior neural approximator and likelihood neural approximator. The proposed framework allows for dynamic parameter estimation over time while quantifying the uncertainty in the estimation. To validate the efficacy of the proposed likelihood-free inference framework, we apply it to a four-story shear frame. The results demonstrate the effectiveness of the proposed framework in estimating time-varying uncertain model parameters.