Structural equation models offer a valuable tool for delineating the complicated interrelationships among multiple variables, including observed and latent variables. Over the last few decades, structural equation models have successfully analyzed complete and right-censored survival data, exemplified by wide applications in psychological, social, or genomic studies. However, the existing methodology for structural equation modeling is not concerned with interval-censored data, a type of coarse survival data arising typically from periodic examinations for the occurrence of asymptomatic disease. The present study aims to fill this gap and provide a flexible semiparametric structural equation modeling framework. A general class of factor-augmented transformation models is proposed to model the interval-censored outcome of interest in the presence of latent risk factors. An expectation-maximization algorithm is subtly designed to conduct the nonparametric maximum likelihood estimation. Furthermore, the asymptotic properties of the proposed estimators are established by leveraging the empirical process theory. The numerical results obtained from extensive simulations and an application to the Alzheimer's disease data set demonstrate the proposed method's empirical performance and practical utility.