I haven't been to Walt Disney World since I had my child, but I've been there roughly a dozen times in my life (most recently four years ago) and stayed in a Disney hotel every time. I agree with the others that I would think you could definitely do Disney without a car seat or car, ever since the introduction of Magical Express. From my knowledge, the buses used for Magical Express are coach buses, and the ones used for getting from your hotel to the parks, etc. are city buses, with no seat belts at all. I've never traveled with my 11 month old on public transit, but I don't see how you would even have the option to restrain your child on a city bus, nor to I know much about the safety of unrestrained children on city buses. If you felt very strongly about wanting to strap your child in everywhere you went, I think you would need to bring your car seat, rent a car, and drive everywhere.
The buses do go to Downtown Disney and the water parks as well as the theme parks. You can get from any point A to any other point B within Disney (hotel, theme park, Downtown Disney, restaurant, etc) using only the internal Disney transit, though you might have to transfer at points. The monorail stops at the Magic Kingdom and Epcot theme parks, as well as the Contemporary, Grand Floridian, and Polynesian (note: all more expensive) resorts - so you could get by on monorail only if you only visited those places. I know there is also a boat that goes between Hollywood Studios and Epcot theme parks, as well as one that goes from the Port Orleans resort and Downtown Disney.
Also, just to throw in my personal opinion, if we ever decide to go to WDW when we have any young kids of baby/toddler age (especially any with a short tolerance for waiting/lines), I would probably consider renting a car and driving ourselves to the parks. That wouldn't be as convenient because you do have to drive/find parking, and you wouldn't get dropped off at the door, but time wise, I actually think it might be a wash. Sometimes you have to wait awhile - sometimes as long as 20 or 30 minutes - for the right Disney bus to come by your hotel, and at the end of the day when the parks close, the buses to take you back to your hotel can be VERY packed and you sometimes have to wait for a number of buses before you get on. Also, if you get on a bus after all the seats are taken, you'll have to stand the whole way back. Standing on a crowded bus with a stroller and a potentially tired/cranky toddler does not sound like fun to me. I'd rather be able to stroll the child back to our car and put him in the comfort of his car seat, even if we had to wait in traffic when leaving the park. Again...I haven't actually been to Disney with a young child, but based on my prior experience and what I now know about traveling with a child, I think it would be preferable to drive ourselves places in the park.
Edit: I just went back and now see your children are older so I think you would be just fine taking only Disney transit - though you know your kids best. I forgot that car seats/boosters are now required for older kids. But I'll leave in my comment about how I think Disney transit may not be compatible with a very young child.