Streaming data routinely generated by social networks, mobile or web applications, e-commerce, and electronic health records present new opportunities to monitor the impact of an intervention on an outcome via causal inference methods. However, most existing causal inference methods have been focused on and applied to static data, that is, a fixed data set in which observations are pooled and stored before performing statistical analysis. There is thus a pressing need to turn static causal inference into online causal learning to support near real-time monitoring of treatment effects. In this paper, we present a framework for online estimation and inference of treatment effects that can incorporate new information as it becomes available without revisiting prior observations. We show that, under mild regularity conditions, the proposed online estimator is asymptotically equivalent to the offline oracle estimator obtained by pooling all data. Our proposal is motivated by the need for near real-time vaccine effectiveness and safety monitoring, and our proposed method is applied to a case study on COVID-19 vaccine safety surveillance.