Hi, Carrie! I am a person who is cold MOST of the time! However, that never happens in WDW in June! Temperatures will be well into the 80's. If you plan to be out very late in the evenings, maybe pack a long sleeve T-shirt for each of your kids, but I really don't think you'll need them (the t's won't take up much room and you'll have them "just in case!")
Speaking of June weather, it is really hot during the day. Plan to get an early start, take an afternoon break, then head back out in the evening. The pool area was recently updated at the Contemporary with a little splash area, perfect for your kids! Take time to enjoy it.
To get to early Extra Magic Hours at Hollywood Studios, simply take a bus. Buses run approximately every 20 minutes, and the bus stop for the Contemporary is just outside the main entrance.
To get to
Cape May Cafe at the Beach Club, I would walk over to the bus stop at the Magic Kingdom (less than a 15 minute walk from Contemporary.) Hop on the Beach Club bus. Once you board the bus, the ride time is about 20 minutes.
Wishing you a magical vacation! What a great way to kick-off summer!