Generally June is spring runoff, but that all depends on the snow pack and how the temperatures go in the spring. A hot spring and you get a quick runoff, a cold spring and the runoff can last a bit longer. Hopefully, you will be there just at the end of the runoff.
For Yellowstone I like Blue Ribbon Flies (West Yellowstone) and Parks Fly Shop in Gardiner, MT, their both great fly shops.
Last year at the end of June we fished the Madison River for a week, primarily using caddis patterns and salmon flies. We primarily fished between the lakes and down river of Quake lake (Reynolds bridge and $3 bridge areas), but the Madison in the park was also fishing really well, as I remember. Just guessing with the snow pack we have at this point, I would expect the Firehole and Gibbon to fish well.
One thing I would recommend if your planning on doing any fishing on your own is getting on the Blue Ribbon Flies web site and sign up for their weekly newsletter. They will report what the local rivers are fishing like and what is working, so you will have a good idea on how to fish there.
You should also be keeping an eye on the USGS water data site, looking at the rivers of interest as you get closer to your trip, that way you will know the status of the spring runoff.
USGS Current Water Data for the Nation
For Jackson, I don't have any recommendations for fly shops or guides, maybe someone here with personal experience will chime in. For rivers you have the Snake, the Gros Ventre (north of Jackson, it crosses the highway), Pacific Creek (north of Moran Junction), the Hoback (south of Jackson) and Granite Creek (again south of Jackson).
