I actually wrote the old Oklahoma Size & Weights Permits system (the one they are using until the new Cambridge System takes over in August) and as part of the design did a survey of existing state permitting systems. Because of the vast disparity between municipalities and their systems/infrastructure, every state I know of has language that requires the trucking company to contact any city/county whose roads they will be transporting on to get approval prior to shipping.
Obviously some companies do not do this but the states are effectively left with this as the only solution as they neither have authority or responsibility for municipally owned roads. My guess is that even IF the states' permit systems could be designed to handle non-state roads (and many of them can) for legal reasons they are unlikely to do so.
As for a county querying a permit system, I don't know of any that do but there is no technical reason why they couldn't. It might take some time/money to develop a way to give the county access but it certainly could be done and it would definitely be helpful for enforcement.