Hi Elizabeth!
You can go to the "Attractions" link, which you can access by clicking here.
Once there, go to the menu option to the left and click on Recommended Ages. Place a check mark on Pre-Schoolers, and the attractions will now filter to all attractions appropriate to your 1 year old. Many of these attractions are slow moving, and will allow you to hold your child in your lap as the ride moves on.
But you're correct, in the parks as a whole there's really no true place babies can just crawl around. There are benches throughout, and of course restaurants with outside seating, but if your child can't walk yet you'll find yourself walking with them near those areas. The crowd levels alone could make crawling a safety risk, even in childhood-friendly areas such as Toontown or Fantasy Land at Magic Kingdom.
You and your little one have a magical time!