Biased data can lead to unfair machine learning models, highlighting the importance of embedding fairness at the beginning of data analysis, particularly during dataset curation and labeling. In response, we propose Falcon, a scalable fair active learning framework. Falcon adopts a data-centric approach that improves machine learning model fairness via strategic sample selection. Given a user-specified group fairness measure, Falcon identifies samples from "target groups" (e.g., (attribute=female, label=positive)) that are the most informative for improving fairness. However, a challenge arises since these target groups are defined using ground truth labels that are not available during sample selection. To handle this, we propose a novel trial-and-error method, where we postpone using a sample if the predicted label is different from the expected one and falls outside the target group. We also observe the trade-off that selecting more informative samples results in higher likelihood of postponing due to undesired label prediction, and the optimal balance varies per dataset. We capture the trade-off between informativeness and postpone rate as policies and propose to automatically select the best policy using adversarial multi-armed bandit methods, given their computational efficiency and theoretical guarantees. Experiments show that Falcon significantly outperforms existing fair active learning approaches in terms of fairness and accuracy and is more efficient. In particular, only Falcon supports a proper trade-off between accuracy and fairness where its maximum fairness score is 1.8-4.5x higher than the second-best results.