Magic Kingdom:
Baby items can be found in Mickey's Emporium on the left side of main street as you are coming in. Also, there are baby snow globes in the jewelry shop on the right side of main street as you come in. But if you are flying you can't get a snow globe. They can't even be checked.
You can get Baby's First Christmas Ornament at the Christmas shop in Liberty Square. They can personalize it too. And you can never have too many 1st ornaments. IMO
The shop at the end of the Pooh ride has lots of Pooh items, clothes, plush etc..
The huge Tent in Toontown has tons of kids and baby items including onesies and toys.
Disney Hollywood Studios
There's a big kid/toy shop on the right. It's the last shop as you come in before you turn onto the street that takes you to Tower of Terror.
There's a Christmas shop around the corner from the Muppets and there is a Muppet shop. I think there were baby clothes in the Muppet shop.
There is a shop outside of the Animation Studios. It's not the shop with all of the art attached to it. It is across from it. Also across from the Playhouse Disney Stage. One part of the shop is open and has princess stuff. The other part of the shop is enclosed and has toys. We got Will a Disney ring stacker and a Disney shape puzzle. You know like the plastic ball that has shapes you can put in the corresponding holes.
Animal Kingdom.
We were pretty much done shopping by this park so not sure where stuff would be here. I know they have toys in the Chester and Hester shop in Dinoland. I'm thinking one of the shops toward the front of the park after the Oasis would have baby clothes.
Epcot:
Also didn't pay too much attention to stuff. There are two shops however, that have toys that are right in front of the World Showcase Lagoon on the Future World side. Basically directly across the water from the American Pavilion.
Also, there are some super cute Nemo onesies at the Living Seas.
Downtown Disney: (Marketplace side)
There is a small baby section at the World of Disney. Go in through the Princess entrance and make a left. Also at the far end of the store there is a big boy section that is completely themed to pirates. Don't think they have baby clothes but they may have some toys.
There is a baby shop on the Marketplace side. It's sort of convoluted so best to look at a map. They have tons of pooh stuff and other stuff. Best selection of baby stuff anywhere on property. They had tons of clothes.
Also, you'll find that just about everywhere there are more girl options than boy. Such is the way of all stores and not just Disney.
You can also buy My First Ears at any of the ears shops. Don't know if they still do it but the Main Street shop at the Magic Kingdom was the only place left on property that embroidered the name by hand.
Happy shopping!