Yep, Veggie has said everything. I'm trying to complete the shipping list AND cook all the recipes, so I need lots of white herbs, and it's very rare and found naturally only in Winter. Hmm, do the badgers bring the white herbs in other seasons?
Yes, the badgers will bring white herbs in any season just as the ducks bring bottles and pirate treasure in any season and rabbits will bring any season flowers all year round.
Just remember that in Winter I also encountered less white herb all season long on Volcano Island when I used the badgers.