Hullo Tony!
Congratulations for so many things...your baby, your nuptials and your holiday in Disney World!! What a great year y'all are having!
You do not need a car seat for the baby on Disney transportation. For taxis and town cars you can request one with a car seat, but it is not required that you have one. They keep some on hand for just this reason and some taxi drivers even keep one in the trunk. If you google 'Disney World Town Car' you'll come up with some reviews on companies that provide town cars and limos for transportation. I used one called Happy Limo years ago and it was a great experience. They brought our luggage to the car, provided a car seat on request, stopped at a grocery store for us, had a clean car and professional driver. Just a great way to start your time in Florida.
I brought my daughter to Disney at the age of 7 weeks and then 7 months. Both times we did bring her car seat that fit into her stroller, but strollers usually have to be folded while on the buses so I ended up using a baby sling a lot. The sling was also handy when I wanted to see an attraction in the Parks that didn't allow the stroller.
I hope that you have a Magical time in Disney World and your marriage is filled with Pixie Dust!