I am not going to comment on whether or not to use seats on a plane, because I always end up on the losing side of the battle.
That being said, do NOT rent carseats. I know you said you didn't want to spend the money, but its not about the money. You don't know what accidents the seats have or have not been in, you may not be able to find the expiration date or where to look for it, you don't know if the straps have been washed or submerged in water, you don't know if the seat has been recalled, etc.
If you are taking any sort of public transportation, 100% bring the seats, no question.
If you are using disney transportation only, you need to decide if your child will be safer in the carseat on the plane, or not.
ETA: We have used the traveling toddler, which is a strap used to attach the carseat to your carry on luggage. It's helpful in that you don;t need to carry the seat, and you can strap the child in the seat and travel through the airport, like you would a stroller. They cost around $15-ish and are well worth it. We used Britax Marathons and it worked wonderfully. We also have 3 Radians, but they fold up flat and you can get the shoulder strap to take it on the plane. Both are good options.
*** Do not check your carseats, you don't know what kind of damage they will do and if you will be able to see that damage or not. A stroller? Yes, I'd check that. But a carseat is the one thing keeping your child off the pavement in a serious car accident, and I wouldn't chance having it damamged in transit and not knowing about it.