In your situation I'd fly and rent a car/van on arrival. You have to remember that with a 4 month old your drive WILL take you longer than the usual 15 hours because you are going to have to stop more often for feedings and changes. I drove from Chicago to PA when DD was about 3 months old and we had to break what would normally be a 12 hour trip into two days because 1) we didn't want DD in a car seat that long in a single day and 2) we knew that hours would be added to the trip because we'd have to stop frequently - I couldn't very well nurse or change DD in a moving car! Granted if your new baby is going to be formula fed or will take a bottle of pumped breast milk that will help but you would still have to make much more frequent stops. If you do decide to drive, I would take two days to drive both ways.
If you do fly please consider buying the new baby a seat and taking a car seat with you. It's the safest way for baby to fly, severe turbulence or a hard landing is no joke.
If it's a 3 hour drive to the airport, you could always drive to the airport the day before and rent a room for the night so you can fly out early the next morning. Many airport hotels have a "park n' fly" rate, you can leave your car parked at the hotel for the whole time you're gone and get a room for a night all for one low rate, you just take the hotel shuttle to/from the airport. Often park 'n fly rate works out to be the same as a week's worth of parking at the airport so it may not cost you any extra.
Strollers and car seats can be gate checked at no fee. Ya know, in case you missed it earlier in this thread!
Now granted I've only flown with one kid and not three. But what we did when DD was 1.5 was that DH and I each had backpacks as our carry-on and DD was in an umbrella stroller (a full sized would have been ok too, either way it's gate checked). We also brought DD's car seat for use on the plane and in the rental car. I bought one of those old fashioned metal luggage carts, the kind popular before wheeled suitcases became common ($20 on
Amazon). DH bungee corded the car seat to the cart, so while I pushed DD in the stroller he could push/pull the car seat. We checked our larger luggage and just had the backpacks. It was super easy to navigate the airport, once the car seat was installed on the plane we could just stow the cart under the seat or in the overhead. In your case you could do the same since the older kids should be able to walk and have small backpacks of their own, or maybe do a double stroller for the baby and the middle child and just the oldest can walk. You can check luggage and car seats that aren't being used at the ticket counter, so you're only dealing with the baby's car seat for the plane when going through the airport (assuming you get baby a seat of course).
I would rent a car because while 100% Disney transport is certainly workable, having your own car can really be much more convenient (especially for water parks/Disney Springs or just a quick get away back to the room if there is a melt down). It also allows you to run to the grocery store and/or eat at less expensive restaurants off property.
For a rental car once in Orlando, if you want to travel a bit lighter you can rent car seats and booster seats from the rental car company. There are additional fees however. You can always bring one or two seats from home and rent the rest to balance cost and convenience.