In this paper, an affine frequency division multiplexing aided grant-free random access (AFDM-GF-RA) system is designed for low Earth orbit (LEO) satellite Internet of Things (SIoT) networks, where users employ the GF-RA to access the satellite concurrently by configuring the AFDM modulation. In order to evaluate the system’s performance, average bit error probability (ABEP) bounds are first derived by Cholesky decomposition, which are verified by simulation results. Furthermore, a block sparse prior expectation propagation detection (BSPEP) is developed for joint active users detection (AUD) and data detection (DD). User activity probability is considered as a prior for joint detection. Concurrently, the accuracy of AUD is further improved by combining the block sparsity characteristics of the user transmission signals. Simulation results show that the proposed detector outperforms the classical oracle least squares (OLS) benchmark and approaches the theoretical ABEP bound at high signal-to-noise ratio (SNR).