I believe there is something at Animal Kingdom but have no clue where everything is. Just check at the Guest Relations to your left right after you go through the turn styles.
At Epcot, the Kidcot stations are at each country and they cost nothing. Your child can start at any country and starts a mask to which he/she will add at additional countries. The stations don't open until 1:00 so plan accordingly. The kids can stay as long as they (or you) want. If it's raining, the stations will be moved inside. You just have to look for them or ask a Cast Member.
You can get a passport and have the Cast Members sign on their respective countries. It's been a while but I think the cost is around $10 or so.