I go to dollar stores, Target's dollar spot, and drug stores before we leave town and pick up whatever cute and cheap Disney-themed stuff they have there. I usually have no trouble scoring Toy Story/Cars/Princesses/TinkerBell stuff like stickers, crayons, little books/colouring books, wipe off boards, candy, and small toys. I also buy the one dollar tubes of glow in the dark bracelets (you can string a couple together to make a necklace too). I stash it all in a shoebox and bring it with us. If it's a long road trip, the kids might get an item or two for good behaviour in the car. Then each day I give each child an item--as their 'treat of the day.' It really helps cut down on souvenir requests. I also give each child a budget to choose one item while we're there (we do our shopping during our visit to Downtown Disney, that way we aren't stopping in at every gift shop during time in the parks, since park time=$$$)
I bring along Disney items that we've picked up on previous trips so I don't have to buy the same items twice (pens, autograph books, lanyards/pins, lighted toys for parades, stuffed animals, Disney character tee shirts, sippy cups...)
Also, I bring LOTS of extra batteries, sunscreen, baby wipes, and a fully stocked first aid/medicine kit. I hate having to pay in-park prices for ibuprofen!!
Before I leave town I do searches for printable restaurant coupons, find out which restaurants offer 'kids eat free' deals (and which night of the week they do). While I'm at it, I program our GPS with addresses of the hotel, restaurants we want to go to, and the closest Wal-Mart or Target Superstore (that way we can stop and get sodas, milk, breakfast stuff to stock the fridge/room without paying on-site prices). I also search for Costco gas stations along the route (since we're members and I find that gas is typically at least 10 cents/gallon cheaper there).
Hmmm, can you tell I'm a bargain hunter??

Hope your family has a great time on their trip