We employ the Holstein model for polarons to investigate the relationship among defects, topology, Coulomb trapping, and polaron delocalization in covalent organic frameworks (COFs). We find that intra-sheet topological connectivity and pi-column density can override disorder-induced deep traps and significantly enhance polaron migration by several orders of magnitude in good agreement with recent experimental observations. The combination of percolation networks and micropores makes trigonal COFs ideally suited for charge transport followed by kagome/tetragonal, and hexagonal structures. By comparing the polaron spectral signatures and coherence numbers of large 3D frameworks having a maximum of 180 coupled chromophores, we show that controlling nanoscale defects and the location of the counter anion is critical for the design of new COF-based materials yielding higher mobilities. Our analysis establishes design strategies for enhanced conductivity in COFs which can be readily generalized to other classes of conductive materials such as metal-organic frameworks and perovskites.