This paper investigates joint device activity detection and channel estimation for grant-free random access in Low-earth orbit (LEO) satellite communications. We consider uplink communications from multiple single-antenna terrestrial users to a LEO satellite equipped with a uniform planar array of multiple antennas, where orthogonal frequency division multiplexing (OFDM) modulation is adopted. To combat the severe Doppler shift, a transmission scheme is proposed, where the discrete prolate spheroidal basis expansion model (DPS-BEM) is introduced to reduce the number of unknown channel parameters. Then the vector approximate message passing (VAMP) algorithm is employed to approximate the minimum mean square error estimation of the channel, and the Markov random field is combined to capture the channel sparsity. Meanwhile, the expectation-maximization (EM) approach is integrated to learn the hyperparameters in priors. Finally, active devices are detected by calculating energy of the estimated channel. Simulation results demonstrate that the proposed method outperforms conventional algorithms in terms of activity error rate and channel estimation precision.