The first time we visited, some 50 years ago, it was just before Labor Day and it was t-shirt weather. We had come down from 100+ temperatures in the Missoula-Superior area. Day we left Yellowstone, stopped overnight in Cody to horrific thunderstorms. Yellowstone had gotten half a foot of snow, and our family in Superior were snowed in for a couple of days, foot to a foot and a half. Then it got warm again.
So I would say September weather in Yellowstone is "changable" as need be your plans. It is really difficult to predict the first and last of winter storms in high country, as the first storm is often well before winter setting in, and the last can be well into what looked like Spring.