The vinegar is supposed to help cut the soap out of their feathers. I've heard of folks putting a cup of vinegar in their rinse cycle instead of using fabric softner because it's excess soap that causes the stiffness of clothing and the vinegar helps remove it.
Has anyone tried the "White Minx" hair color rinse on white chickens? We used to use it on white manes and tails and it took the yellow right out. Put too much on, though and you get purple! Since hair and feathers are basically made from the same stuff, I bet it would work. I THINK they still make it.
Just to clear this up, one doesn't actually bathe a bird with ACV. You just put a little of it (about three tablespoons per bucket) of it (or white vinegar) into the rinse water to cut the soap suds from the wash cycle, as it were.
p.s. Thanks Dustin. (Will be working more on your family tree today, btw.)