The network behind these phones is definitely Sprint. I don't think they are up and running yet, even if you can place an order by now.
Just a word of caution: Since this is a brand new service using brand new phones, you may want to wait a bit until the bugs get worked out before signing any long-term contracts. There are also similar-style phones and plans coming very soon from other carriers, so if you are happier with a different provider, just hang on and you can get the same kinds of child safety features from the network you already like.
I love Disney and I wish my phone had more Disney features like ringtones, but I'm not willing to put up with bad coverage or bad customer service.