We have traveled to Sea World several ways. We took the bus one year and while it was cheap (less than $10 for 5 of us each way), it took a long time due to the stops and I would not take the bus again if I had an option. We took a taxi once (before Uber and Lyft), and it was very simple, easy, and not to expensive. And then we have had rental cars which was also fairly easy).
What type of car seats do you need? If boosters, we traveled with a seat called Bubblebum. Uber Car Seat in Orlando will provide one car seat, but if both are in car seats, that wouldn't work for you unless you brought an extra one (and stored it at a locker at Sea World).
The next easiest I would say would be to rent a car with car seats. You could pick up and return at the Orlando Car Care Center (or from a hotel in the Disney Springs area). If you are wanting to get an early start, I would pick the car up the night before.