The flourishing low-Earth orbit (LEO) constellation communication network provides a promising solution for seamless coverage services to Internet-of-Things (IoT) terminals. However, confronted with massive connectivity and rapid variation of terrestrial-satellite link (TSL), the traditional grant-free random-access schemes always fail to match this scenario. In this paper, a new non-orthogonal multiple-access (NOMA) transmission protocol that incorporates orthogonal time frequency space (OTFS) modulation is proposed to solve these problems. Furthermore, we propose a two-stages joint active user detection and channel estimation scheme based on the training sequences aided OTFS data frame structure. Specifically, in the first stage, with the aid of training sequences, we perform active user detection and coarse channel estimation by recovering the sparse sampled channel vectors. And then, we develop a parametric approach to facilitate more accurate result of channel estimation with the previously recovered sampled channel vectors according to the inherent characteristics of TSL channel. Simulation results demonstrate the superiority of the proposed method in this kind of high-mobility scenario in the end.