You can defintely do 5-days for under $2000 -- you just need to plan carefully and be prepared to stick to your budget.
Stay onsite in the All-Stars; yes, you can save a dollar by staying off-site, but you're losing value. By staying onsite you have "the magic" 24 hours a day, and you'll do much more; thus, a greater value. Wishing you could afford a suite? Bring along a small $20 tent, and your kids can have "their own room" within your room; kids don't care about hotel rooms. Onsite you can also use Disney's transportation, which is quick and easy -- and free. Read mousesavers.com for codes.
All-Stars are usually available in the $75 range, so let's assume 5 nights x 75 = 375. Given that you're visiting in early December, you may very well do better than this!
You're staying five nights, but you don't need five day tickets. You probably need 3-day passes; the difference in 3-day and 5-day passes is significant. The first day you will be traveling, and it's foolish to pay full-price for tickets while you're on the plane for most of the day. I'd count on using the first day to check into the hotel, swim, and enjoy a nice character dinner or go to Downtown Disney (free). In fact, you can do quite a few things at Disney for free. Ditto for the last day: there's no need to pay for an expensive ticket when you'll be spending most of your day packing and leaving. Instead, let the kids have one last swim at All-Stars, then go to a nice character breakfast before leaving. If I were you, I wouldn't pay for the plusses; kids as young as yours will enjoy the hotel pool just as much as they'd enjoy the water parks, and December weather can be iffy. I also wouldn't pay extra for the hoppers; Judging from their ages, your kids will probably enjoy Magic Kingdom and Animal Kingdom most.
3-day passes: 2 adults @ 181 each + 2 kids @ 144 = 325 for tickets (source: Ticketmania.com). You won't find significant discounts on tickets anywhere, and buying used ones from ebay is very dangerous; No choices -- you'll just have to pay these high prices for tickets.
So you have two huge expenses done: hotel and tickets, and you've only spent 700.
Food will be your next big expense: Bring snacks and breakfast for the hotel room -- that saves a ton right there. There's Deb's website where you can check out menus ahead of time; Can someone post the address? I forgot it. That'll allow you to choose your meals carefully. Consider having the kids share a kids' meal (with an extra drink) -- they're big enough, and you can always order more if you're still hungry. We usually spend about $100/day for our family, which is enough for one sit-down and one counter-service meal each day. That'd add $500, bringing your total up to 1200.
Souveniers -- just say no -- seriously. I do, and my kids don't even care; they understand that they get to travel much more than other kids, and they're fine with not taking home a tee-shirt or a stuffed animal from every place they visit. Or, if you're not as cheap as I am, then set a dollar figure and explain that's it. Many people with young children buy Disney-themed items ahead of time (at
Walmart, etc.) and give them to the kids at WDW.
That leaves only airfare and other transportation. I rarely fly anywhere, so I'm not much help here, but I believe you should be able to fly the family for the remaining $800. If you're staying on-site, you'll be fine without a rental car -- plus you'll avoid the hassel of getting lost and finding parking spaces (don't underestimate how big Disney is). You will need to budget a few dollars for parking at your home airport and a shuttle service /taxi to get you from the airport to WDW. Still, that should easily fit into the $800 that's left for transportation.