First of all... CONGRATS!
My youngest DD was 18 mths on our first trip, so I can't help ou with the logistics. Do bring your stroller right to the airplane door so you can have it immediately at both airports. As for carseats, the buses did not have seatbelts.
Each baby is different -- at 6 mths, they may be in the infant carrier still (up to 20 pounds) in which case, I would definitely bring it for ease and comfort of transport (especially if it can click onto a stroller). My first DD didn't hit 20 pounds until after her first birthday. My second DD hit 20 pounds before 4 months! Ya never know.

However, if they're already in a regular babyseat ... I'm not sure. Hopefully, Sean and Minnie or Realfam can answer baby questions.

Again ... congrats ... and definitely take it slow at WDW and savour it!