Although it is better known for its Summer and Winter activities, Autumn can also be a very appealing time to visit America's oldest national park. The crowds of tourists are much smaller in Yellowstone at this time of the year and the wildlife can be much easier to spot as well.
Typically the prime time for viewing color in Yellowstone National Park ranges from the last week in September through the first week in October. Mid-September can often present some nice color in the higher elevations. Keep in mind that lodging options inside the park start closing down around the middle part of September, so you will certainly want to plan ahead and make sure that you can still find available lodging that will fit into your plans. You might have an easier time finding available lodging if you look at the lodging options just outside the park's gateways. There are many lodging choices at each of the parks gates, and you can find more information about each location here.
Keep in mind also that Fall weather is unpredictable in the park. Roads may be closed temporarily by snow or other potentially dangerous weather conditions, and snow tires or chains may be required.