How to remove/prevent standing water at campsite???

If none of the other suggestions work for you (and they are good suggestions), one other idea is to drop a couple of 2x4's across the low area and nail some 1x2's across the top. That will make a wood open deck that will let some light to the grass underneath but will let you walk across the bog and not get wet. When you don't need it any more, you have firewood. Total investment would be under $10 so it is not an expensive solution.