The requirement is due to the federal flood insurance program. FEMA had to assemble the flood plain maps which were put together by the states. Because ANY new structure in the flood plain has the potential to displace flood water, local authorities are required by federal regulations to implement a permitting process for such structures.
FEMA doesn't issue permits. That's the responsibility of the local permiting authority. In rural West Virginia areas it's the responsibility of the county commissioners. In rural WV I can build anything I want without a permit if it's not in the 100 year flood plain. The county commissioners have absolutely no say. You might get way with building a bridge without a permit but based on a Army Corps of Engineers meeting I attended two days ago, the flood plain maps are to be redone in the near future.
If during that process it's discovered that someone has an unpermitted structure there may be reprecussions. Those would probably depend on the local situation and whether or not there was a prick on the local government payroll who wanted to make a public example of someone. This is a situation in which it will make life much simpler to dot the i's and cross the t's beforehand. If the feds weren't paying for flood disaster losses, the requirement wouldn't exist.
I'm just the messager here. I'm preparing to build a multiple span bridge one span of which will be 149' to pass completely over a creek. The bridge itself will be above the 100 year flood elevation but the center pier will be within the floodplain. I've talked to the Corps, FEMA and the local folks. That's why I'm familiar with the requirements.