You may be a little late to find any accommodations for June, even on the outside near the park. The closest city/town to fly into in Cody. And, there is a Holiday Inn there. I would recommend contacting them immediately. I assume there must be day tours into the park there. Or, of course, you could rent a car. I would suggest you call the Holiday Inn in Cody (or perhaps there is a Chamber of Commerce there that could help with both room availability and info on tours).
If you do go to Cody, I heartily recommend you visit the outstanding western museum there.
I also strongly recommend that you spend part of your visit in the Grand Tetons, which are absolutely magnficent.

Adding to my previous post: Cody is just east of Yellowstone.
The next closest airport is Jackson, which is south of the Grand Tetons, which, in turn, is south of Yellowstone.
The Jackson airport probably has more planes in and out than Cody.

Pati, my family made this trip 2 years ago in June. We flew into Jackson, rented a car and drove up to West Yellowstone. I highly recommend the West Yellowstone Convention Center (a Holiday Inn). If you have children, request a family suite.(Info @ www.yellowstone-conf-hotel.com) We also stayed at one of the "in-park" hotels. It was very cramped and damp. I'd go back to West Yellowstone any day! Check out www.Yellostone-Natl-Park.Com and www.gorp.com for great info. on the park. Be prepared for any kind of weather. We wore shorts the first 2 days, and got 10 inches of snow the 4th day. Have a great trip!

I think Delta has flights into Jackson, WY. Another airport that has a slightly wider selection of flights is Billings Montana. It is not that far from Yellowstone, IF the Beartooth Highway is open. For early June, that is a big IF.
Other than that Missoula Montana is fairly close.
It is not much farther than Jackson; and it might be the closest place to stay!!
Access the web site for West Yellowstone Montana and see if a motel there has rooms left.
Cabins and lodges in Yellowstone are full unless there was a major cancellation by a tour company.

Cody and Jackson aside, your next best bet from the north is Bozeman, MT, a 2 our drive to Yellowstone. I've flown Delta and Northwest into Bozeman. It's much closer than Missoula or Billings. We've made that route in December, so it would definately be open in June.

If you fly in from the north I would stay in West Yellowstone. The Stage Coach Inn is nice and very clean. We stayed at the Best Western Executive Inn. Make contact now as they will book up. We managed to get in with only a month notice last August.

My favorite was the Sportsman's High. It's a wonderful Bed and Breakfast that also has modern rental cabins (some with hottubs). They are located just west of W. Yellowstone. We've stayed there twice and really enjoy the breakfast! If you are lucky you might see moose in the morning. Check them out at www.bedandbreakfast.com/bbc/p210383.asp

OOPPS. I mispoke when I said Missoula. Wrong university!! Bozeman is much closer. In fact we spend the night there last year on our way to Yellowstone. I just pushed the wrong brain cell button and Missoula popped out.

One reason for Billings is that the drive over the Beartooths is very spectacular. It is worth a detour just to see it.

As for scenery I have two "must do" attractions. Check the schedule for either Castle Geyser or Grand Geyser
and wait for it to erupt. Grand Geyser by the light of a full moon is dramatic.
The other one is the falls of the Yellowstone. Spectacular and beautiful as well.

from a montana native. Second the recommendations on Bozeman for airport access and West Yellowstone as a place to stay. Plan lots of time as traffic is slow in Yellowstone with buffalo and/or bear "jams"