The large number of users and the long propagation delay of communication satellites make it difficult to deploy a random access scheme that performs well in terrestrial networks. The aim of this paper is to demonstrate that a relay architecture with unmanned aerial vehicles (UAV) between the ground and satellites can facilitate satellite access with a low overhead. We consider a TDMA-based two-level relay structure, in which users transmit data to UAVs through a random access scheme, and the UAVs forward it to the satellite with either a coordinated or uncoordinated access scheme. We show that the coordinated access for the UAV-to-satellite links can substantially improve the throughput and delay performance. We conclude that a proper scheme for resource allocation between ground-to-UAVs and UAVs-to-satellite is critical to optimize the relay network, which includes the UAV altitude, and should be based on traffic demand and the number of UAVs deployed.