Mornin'.
Weather in YNP is unpredictable any time of year. According to YNP web, it can and does snow in YNP any month of the year. For each of the last 2 years, we were there in June and both years we had overnight temps. of 34 degrees and freeze warnings. Once when we out there near the end of June, it snowed enough for my kids to build a snowman, and we were officially closed off in our over-night area and not permitted to leave by Rangers until roads were cleared.
Last year a good friend of mine planned a mid-September visit to YNP & Jackson. I told him about the weather and he planned for the possibility of snow. Good thing, because they were not able to do everything they wanted because of an early snow closing some roads and passes. But, they had a plan B.
So bottom line from this periodic visitor to YNP, go and enjoy. But, plan for the worse and hope for the best. Cold weather and snow are probable but not a sure thing.
The weather in Jackson is similar to YNP, so the advice is the same.
Have a great time!
Bob