We have done Disney several times on NYE and it is a packed house at the parks. 2000 was the worst I ever saw, though.
So prepare for the crowds of people, BUT I have always found the least stressful to be the Boardwalk or *Hollywood Studios (*Dine there but get out before the crowds (11:00 pm or so) and head to Boardwalk to watch fireworks with glass of champs from one of the restaurant bars).
If they do not want to head to a park, what you can do is what others have said, make dinner reservations at MK resorts restaurants or the Boardwalk resorts. Linger there till midnight and watch the fireworks.
I prefer Boardwalk for reasons mentioned above and you can actually take a lovely stroll after dinner, shop, grab ice cream/dessert and then watch the fireworks.
Do not forget about the restaurants in the Dolphin and Swan hotels. Shula's and Il Mulino may have availability earlier on, Bluezoo has some later time slots. You get the Disney experience without the "crushing" crowds, but there will be crowds.