Symbiotic radio (SR) is a promising technology to alleviate the tremendous spectrum demand from massive Internet of Things (IoT) devices. We notice that the throughput of SR systems with multiple passive IoT devices highly relies on how these devices are scheduled, i.e., which backscatter device (BD) transmits passively when there is an active ambient terminal. Due to the dynamics of wireless channels and the random access of terminals in ambient networks, the scheduling of BDs needs to be adjusted adaptively while ensuring fairness among them. In this work, we design a transmission protocol for an SR network where an ambient system adopts a contention-based channel access protocol, and formulate a multi-BD scheduling problem considering fairness among BDs. To solve this problem, we propose a soft actor-critic-based learning algorithm. Simulation results have illustrated the effectiveness of the proposed algorithm. They have also demonstrated a throughput-fairness tradeoff, and the impact caused by the timeliness of channel state information.