What is a free and easy-to-use website for organizing an online correspondence chess competition for kids?
You may want to use the echecsemail website. You can create round-robin tournaments for up to 9 players. For bigger player-pools, you'll have to split them in groups (or maybe talk to the admins).
They also offer the possibility of choosing an alternative starting position (for opening practice purposes mainly)
I hope those kids speak French, though!
https://www.chess-server.net/en/ is another option. 100% free!!! 10-20 different languages are available. The director, Ortwin Paetzold, is extremely helpful. Just contact him.