I think you can do it!
My biggest suggestion is to stay offsite. We stayed in a wonderful 3 BR, 2 bath Windsor Hills condo last November (14 - 21) for $299 for the week! The total (after taxes and a $75 cleaning fee) was about $420, or $60/night! (They have a free shuttle to the parks so you don't have to pay for parking.)
If you have a kitchen (like we did) you can save a LOT of money on food. I know I don't like to cook on vacation, but we ate a quick breakfast (cereal, yogurt, fruit) at our condo every day. We also brought snacks/drinks into the parks every day and lunches several days. When we ate meals in the parks, it was mainly counter serve meals. We chose one character meal for the week (my kids chose Chef Mickey's) and it was special since it was the only one we did. But my kids would gladly do without the character meal if we needed to save money somewhere.
Drive instead of fly. I know you will spend money on gas, tolls, and maybe a (hopefully cheap) hotel on the trip, but you will save a lot on airfare and also a rental car. Bring as much food as you can for the drive and for the trip (breakfsts, snacks, drinks).
Get tickets through undercovertourist.com and use the mousesavers link. You won't save a ton, but they are definitely cheaper than getting them at the gate and I have had great luck with them. I agree with the PPs, don't get park hoppers. We choose one park per day and have always been happy with that.
I know this can be hard, but stay out of the gift shops! My kids love the (free) Kidcot stations at EPCOT and making the masks. There was a thread somewhere a while ago about all of the fun free stuff to do at Disney World. Maybe you could find it to get some good tips.
Some quick numbers for one week (not sure how long you will be staying):
MYW tickets for seven days (assuming one under 3, three 3-9, and 2 adults) = $1,100 (
undercover tourist)
Lodging (see above) = $420
Food for seven days at $80/day (assuming eating breakfast in, bringing lunch, snacks, & drinks to parks, and counter serve dinners) = $560
TOTAL = $2080
This leaves $420 for the drive down and back (not sure how much gas will cost) and any "extras".
Good luck!!