How nice of you to be planning a trip for your grandchildren. WDW is a super place for grandparents and grandchildren to spend special time together.
The Caribbean Beach resort is beautiful. I would not choose this resort for a June visit because the main pool at Port Royale is going through an extensive rehab. Once completed, it’ll be a fantastic place for families to swim and play. Caribbean Beach has sandy beaches and playgrounds for the kids. Rooms are a good size and the quaintness of the buildings, courtyards and paths detracts from the vastness of the resort.
Since you are traveling with 4 children, I would recommend
Port Orleans Riverside. Room occupancy at moderate resorts is 4 people plus an infant in a crib. If this doesn’t work for your family, Port Orleans Riverside allows 5 people plus an infant in a crib. You would need to request a room with a trundle bed in the Alligator Bayou section.
Coronado Springs would probably be too much walking as the resort is quite vast. That same philosophy would hold true for Caribbean Beach however for an extra fee you could request a premium room to put you closer to the resort’s amenities.
Port Orleans French Quarter would be nice as it is a small resort and is one of my favorites. The pool may only appeal to really young children but I’ve found this to be a great resort when traveling with my son’s grandparents.