Deploying autonomous systems in safety critical settings necessitates methods to verify their safety properties. This is challenging because real-world systems may be subject to disturbances that affect their performance, but are unknown a priori. This work develops a safety-verification strategy wherein data is collected online and incorporated into a reachability analysis approach to check in real-time that the system avoids dangerous regions of the state space. Specifically, we employ an optimization-based moving horizon estimator (MHE) to characterize the disturbance affecting the system, which is incorporated into an online reachability calculation. Reachable sets are calculated using a computational graph analysis tool to predict the possible future states of the system and verify that they satisfy safety constraints. We include theoretical arguments proving our approach generates reachable sets that bound the future states of the system, as well as numerical results demonstrating how it can be used for safety verification. Finally, we present results from hardware experiments demonstrating our approach’s ability to perform online reachability calculations for an unmanned surface vehicle subject to currents and actuator failures.