Cut the potatoes into small bite-size pieces that are similar size (no need to peel the potatoes, but you can if you prefer). Add them to a large pot (don't salt the water) and ensure that they're covered by 1" of water. Bring the potatoes to a boil over medium-high heat, then reduce the heat and let them simmer until they're tender when pierced by a knife.
While the potatoes cook, add the dressing ingredients to a small bowl and mix together until smooth. Refrigerate the dressing until needed.
Once the potatoes are cooked, drain them thoroughly and then return them to the pot they were cooked in. Drizzle them with the white vinegar and toss gently with 2 spoons. Let them cool for at least 20 minutes.
While the potatoes are cooling, prep the celery, onions, and pickles, and add them to a large salad bowl.
After the potatoes have cooled for the 20 minutes (they will still be warm but not so hot that the salad dressing will separate), add them to the salad bowl along with the dressing. Toss gently until coated. I like to sprinkle a little extra Tony's Creole Seasoning on top.
Serve immediately or cover the bowl with plastic wrap and refrigerate to let the flavors meld a bit more prior to serving.