September is a good month to be in Yellowstone. For wildlife image opportunities, I would try to be there the last two weeks of the month. Early September can still be summer-like, with no trace of rutting activity from either the moose or the elk. By the time mid/late Sept rolls around, both the elk and the moose will have definitely shed their velvet, and rutting action will be well underway. This is really the type of behavior that makes for more interesting imagery.
Another benefit to later in the month is that the Whitebark Pine nuts will be ready for the bears to eat. This brings both black and grizzly bears down to the areas where they are most readily seen and photographed - particularly, along the roads in the Dunraven Pass area.
Be sure to keep an eye open for opportunities to photograph the littler critters such as Pikas, Snowshoe Hares, and the like. September is a great time for Pika photography, as their feeding and cacheing activity is at its absolute peak.
I hope you enjoy looking forward to your trip, researching the area, and making detailed plans - that's half the fun!
"Your" and "you're" are different words with completely different meanings - please use the correct one.
"They're", "their", and "there" are different words with completely different meanings - please use the correct one.
"Fare" and "fair" are different words with completely different meanings - please use the correct one. The proper expression is "moot point", NOT "mute point".