Welcome to the disboards. You've come to a great spot for opinions. There are 60,000+ members each with a different one, lol!
With small children I would choose to stay at a monorail resort as you will undoubtedly be at Magic Kingdom a lot. We've stayed at the Contemporary and our next trip is planned for the Polynesian which we have loved touring every trip. The Contemporary is usually a bit cheaper than GF or Poly. Another option for closeness in the Wilderness Lodge. There rooms are set for 4 but you must have one under three in a crib if your oldest is only 4yrs old.
Our three boys are now school-aged and we are sort of stuck taking trips during summer and school breaks. When ours were younger we really enjoyed vacationing in the off season. If you are set on June then go with it but remember the summer is pretty crowded, Florida in the summer is obnoxiously hot and it rains nearly every day. On the plus side the parks are generally open longer hours so you can do more. Our last trip was in June and we said never again after it rained every single day. Typically the rain lasted 2-3 hrs but we found it disappointing. Our upcoming trip is for Christmas and we will have to fight the crowds. Seems there is always a trade-off when you have schoolagers.
Hopefully, summer discounts will be coming out soon. The current discounts run thru mid-April.
Good luck!