It really depends on you and your kids, and no one else can guess that.
With that said, it's "doable"- if you think you and your children can handle it. We drove 20hrs straight just a little over a month ago- with children very close to the ages of yours (they were 2, just turned 5, and almost 9yrs old). We just made lots of stops (not scheduled, just whenever we needed to -to eat, get a snack, change diapers, restroom breaks, stretch our legs, whatever.)
There are also things you can do that make the trip more enjoyable for your children. We had a small tv in the car with a vcr so they could watch Disney tapes the entire drive. We brought toys (some that were new to them to keep their attention- but inexpensive ones), things to color (we bought the marker/paper sets that the markers can't write on anything BUT the paper- so we didn't have to worry about everything getting covered in marker LOL), snacks and drinks, etc.
However, this was only "doable" for us because.... I'm a night owl and driving at night wasn't dangerous for us. That is when I'm most "awake". If we were both the types that had set bedtimes and always slept 8hrs at night (for example) then one of us would have been driving while very tired and that's not a good idea- ever.. but especially when you have your entire family with you! So I drove at night, my husband drove during the day. This worked well on the driving- when I got tired after driving that night (getting tired around the time I normally go to sleep around 4-6am) then I just made a "pit stop" for restrooms/etc and my husband took over (he's used to getting up early for work).
Now I will admit we probably wouldn't have been as rested as possible to do the parks that first day. We left in the early afternoon (husband driving, then me driving at night, then him finishing up the drive in the morning) and arriving near Disney around 9am. We didn't do the parks that first day though. We did Wonderworks which was fun for the kids but WAY less walking for all of us and not as tiring as trying to cram a day in the parks in that day. We did wonderworks, checked into the hotel in the afternoon having a semi-late lunch at a Golden Corral there, then went back to Wonderworks again- then did the dinner "magic" show there. It was extremely fun for the kids (my son still asks to go back to the "upside down house" even though it was the first thing we did in a 9 day vacation!) yet it didn't tire us out. We didn't hit the parks until the next day.