When I was little my Mum painted my face with Caran D'ache Neocolor II Watersoluble pastels. I'm not sure if they're vegan, but since they're not technically face paint I doubt they've been tested on animals.
I never got a rash or had a reaction or anything, but I'd do a patch test on your own child just to be safe. They are probably kind of expensive, but one small box lasted us my entire childhood. Then I found it again as an older teen and my favourite colours were only 3/4 used.
You could look in an art supply store, and see if they list ingredients at all? Or email them? Here's their website.
Mix cornstarch and flour together in a bowl. Gradually stir in Karo syrup and water until smooth. Divide mixture into individual containers as needed and tint each one with the food coloring of your choice. Leave one batch untinted.