I usually go to client's homes to shoot babies & kids...but I don't have a permanent studio space. I like the 'service' aspect of going to their homes, where they & the kids are comfortable...but man, it's a lot of work to bring all my gear and set up a studio in someone's home. Not to mention that space is often limited.
I don't have much for kid props, but I did build a 'baby stand'. It's like a sidewalk sandwich board, two boards, a hing and a chain...but I build an adjustable shelf on one side. It's really only good for head shots of the babies, and they have to be old enough to hold their head up.
I've also used a 'Bumbo' foam chair to hold babies in a sitting position.