Kasho Hyoka Izuku

Aizawa does not approve of Ghost's actions on vigilantism. But he did not try and capture the vigilante either. For many reasons.

His main reason, Ghost was a child.

The other reason, Ghost has a habit of ghosting any hero he encounters. The pattern? All of the heroes have tried to capture him using their quirks one way or another. So no, Aizawa was not keen on hunting down children anytime soon. Even vigilante children.

He did not expect to adopt one though. He blames his bleeding heart.