There are a lot of areas.
1) Both Baby Care Centers don't mind if you hang out for a while. Get to know the staff - I know we've made them part of our personal family...
2) At GCH, there are several small seating areas on the odd numbered floors (or is it the evens) in the corner of the hotel that sits over World of Disney - near the Storyteller's/Napa elevators. Those are really nice and there is an outdoor patio/balcony on the 5th floor right off the lobby atrium. Other nice places are the Napa Rose lounge - have a drink or appetizer/dessert and let her crawl - we've done that so many times...
3) The Animation Building lobby in DCA. Best place to chill, nurse, crawl, dance, and whatever - the large movie screens add to the ambiance.
4) Upper level inside Innoventions. Large spaces not filled with attractions. Quieter than the rest of the park, since people kind of run through there and not stop to play. You can take turns watching the baby play and looking at exhibits in the Dream Home and such.
Since folks have different levels of what they allow their child to crawl on, I'm going to stop here and tell you there are some other areas that have more space, but are outdoors textures that are not grass. Some kids like them, some don't. Some parents like them, some don't. Some prefer not to let the small ones crawl near the water areas, some strip them down to a diaper and hold their hands to help them walk in the sprinkles. Depends on what type of parenting you prefer!
Hope that helps!