The increased demand for data privacy and security in machine learning (ML) applications has put impetus on effective edge training on Internet-of-Things (IoT) nodes. Edge training aims to leverage speed, energy efficiency and adaptability within the resource constraints of the nodes. Deploying and training Deep Neural Networks (DNNs)-based models at the edge, although accurate, posit significant challenges from the back-propagation algorithm's complexity, bit precision trade-offs, and heterogeneity of DNN layers. This paper presents a Dynamic Tsetlin Machine (DTM) training accelerator as an alternative to DNN implementations. DTM utilizes logic-based on-chip inference with finite-state automata-driven learning within the same Field Programmable Gate Array (FPGA) package. Underpinned on the Vanilla and Coalesced Tsetlin Machine algorithms, the dynamic aspect of the accelerator design allows for a run-time reconfiguration targeting different datasets, model architectures, and model sizes without resynthesis. This makes the DTM suitable for targeting multivariate sensor-based edge tasks. Compared to DNNs, DTM trains with fewer multiply-accumulates, devoid of derivative computation. It is a data-centric ML algorithm that learns by aligning Tsetlin automata with input data to form logical propositions enabling efficient Look-up-Table (LUT) mapping and frugal Block RAM usage in FPGA training implementations. The proposed accelerator offers 2.54x more Giga operations per second per Watt (GOP/s per W) and uses 6x less power than the next-best comparable design.