Here’s two companies that offer vegan shoes for children:
https://www.jambukd.com/shop.html?featured_icons=236
https://www.mapshoes.com/shop.html?featured_icons=236
Note: These companies also carry non-vegan shoes.
VRG also has an online guide to leather alternatives including shoes, bags, and much more for adults and children here: VRG’s Guide to Nonleather Shoes, Bags, and More