Real-world financial documents contain information on various financial instruments, but they are presented in complex, multi-page, tabular formats. This paper proposes TASER (Table Agents for Schema-guided Extraction and Recommendation), an agent-based table extraction system that continuously learns to extract this unstructured data into a normalized format. TASER performs table detection, classification, extraction, and recommendation, and recommends schema modifications through a Recommender Agent, determining the final result. TASER outperforms Table Transformer by 10.1%, and with large batch sizes, we observed a 104.3% increase in actual schema recommendations and a 9.8% increase in extracted assets. To train TASER, we manually labeled 22,584 pages, 3,213 tables, and $731.6 billion in assets. We made the TASERTab dataset publicly available, making real-world financial tables accessible to the research community.