With 4 kids, forget about staying onsite. Six people crowded in one room at Port Orleans will be a complete miserable experience, and will cost waaaaaayyy more than you need to spend. You are travelling a long way, you don't want to be cramped and unhappy when you get there. You will have a vehicle anyway so you might as well stay offsite in nicer accomodations and be more comfortable.
Check out
www.skyauction.com. Go to the "search" function and type in keyword "ready". That will bring up the Ready, Set, Go auctions. Select one of the auctions that come up (usually there are 3 current auctions) and scroll down about 1/3 of the page until you see "Click here to see where your next vacation will be". Then you can type in your dates, location, and type of accomodation (I suggest min. 2 bdrm). At the moment, there is nothing coming up for your dates in June, but there are several good ones for Sept. If you go with skyauction you can expect to pay somewhere between $350-$400 for the entire week, and will have a full kitchen for breakfasts and snacks, and in suite laundry.
You can also check out
www.floridasunvacationhomes.com, and will find excellent prices for 2 & 3bdrm condos and townhouses for your dates in June for under $100. per night. Most of these are located in Windsor Hills which is something like 4 mins from parking lot to the park entrance at Animal Kingdom. Obviously, WDW is a big place so the other parks are further than that.
The benefit of staying offsite in a condo, aside from the space, is that you will save money by being able to pack cereal and snacks to bring with you in the van so the kids can have breakfast in the suite. You say you don't want to cook, which I totally understand, but you will need to have things for the baby anyway and pouring a bowl of cereal or making some toast isn't really cooking. You would be able to have milk, juice boxes and fresh fruit for the kids. I guarantee that dragging 4 kids out of the hotel room to get breakfast every morning would get old real quick! Being able to throw in a load of laundry would mean packing less clothes for everybody, and you could take more snacks or a case of water.
We never stay onsite, always have a 2bdrm condo (and I only have 2 kids!). Trust me, you won't miss any magic!