I would not use a car seat attached to a stroller for WDW.
We actually tried this a few times during our trips in March and April and it was too hot and too hard - especially folding the stroller for the buses.
Congratulations on your first!
We had the best luck with an easy to fold stroller (umbrella or city mini) and a baby carrier. We'd carry or wear the baby into the restaurant, and sometimes would bounce and shush until she fell asleep, and then we'd have a nice leisurely dinner! Our DD was 3 months on our march trip and almost 4 months on our April trip. She was generally pretty easy to tote around for dinners, We did with with our older DD in Europe when she was a newborn too - stuck her in the carrier, walked around until she fell asleep, laid a napkin over her head, and had a nice dinner. (Note: Whoever has the baby in the carrier should not have soup for dinner

)
For 4 months, I'd bring or rent something like a city mini that has a 1hand fold but is also reclining and has a good sunshade, so baby can nap while you are out at the parks, and also bring the carrier because it is going to be easier on transportation.
Also - I know you didn't ask this, but I'd strongly suggest babybanz or some other hearing/ear protection for your baby. On this last trip our older daughter wanted to see ALL the fireworks, and it was past the baby's bedtime. We would pop the baby in the carrier (usually she'd be asleep since dinner), put the headphones on her, and she would sleep through fireworks too. It was awesome. I'm so glad we took those trips when I was on maternity leave!