Hi Nicole!
I loved visiting Disney World when my daughter was an infant! I brought her at 7 weeks and then at 7 months.
What made visiting the Parks with my baby easy were Disney's Baby Care Centers. Each Park has a center that is a lovely clean and quiet place for you and baby to cool down and take care of things like diaper changing and feeding. Chairs are provided for nursing and microwaves are available for warming up bottles. When I was there they also had diapers and other essentials available.
When you get to the Park, pick up a map and look for the -pacifier- to locate the Baby Care Center.
We brought her super comfy stroller, but I also brought a sling. I used the sling for those areas and rides where we couldn't bring a stroller...that way she was safe and I still had two hands to help my then toddler son, too. A way to shade her from the sun will be important in June. We used a beach towel to shield our little girl and the towel was then used to dry my toddler off when he got in the fountains, so it was a double plus.
Be sure to bring a change of clothes for her! We had one memorable experience in line for the Pirates of the Caribbean ride that had me sprinting to the restroom holding my baby at arms length and yelling for people to move out of the way if they knew what was good for them!! A change of clothing was super important that day...as was bringing plastic bags to wrap soiled items up to contain them!
I hope you have a wonderful time!