I keep finding icings with egg yolks that you use before the cookies are baked. I don't want to use anything with egg in it, because my kids like to eat the icing. I am also wanting to find something that they can use a paint brush with after the cookies are baked.
I think the icing have egg whites but if you prefer not to use egg white or meringue powder you could probably try marshmallow fondant or normal fondant. It is perfect to paint on and you can cut the fondant out while the cookies are cooking with the same cutter and the kids can paint with food colour on top of the icing when they are cool. HTH