Linda,
A truly great question.
So there are play areas, like the Honey I Shrunk the Kids Movie Set Adventure area at Hollywood Studios that
sound ideal. And while very kid-friendly, I would never let the 1-year olds loose there. Very safe environment, but other kids are running free, and you are guaranteed a bumped head crying fest there.
Many other "exploration" areas like the pavilions of the Epcot World Showcase, Tom Sawyer's Island, and other similar ones are either not safely laid out for such youngsters, or have too many people in them to feel like they won't get bumped over.
So what to do?
Well I like some of the suggestions at this
Walt Disney World website page for parent with "little ones". The maps do specifically identify areas where you CAN let your kids loose, like the Toontown Park in the Magic Kingdom, and parts of the Boneyard in the Animal Kingdom. These WOULD be good places to do just what you describe. Just know that more active playground areas for slightly older kids are nearby, so keep an eye out nevertheless.
I hope that this helps? Have fun on the trip!
Have a magical vacation!