It can be mitigated by setting up protections on your home network but phones come with Internet access through the telephone network which is outside of your home network. There is probably a way to control it though.
I image the simplest way to control it is to force all access through your home wifi network and not include your son or daughter on your dataplan. They really don't need Internet access on their phone anyway.
There isn’t any way to buy a phone plan that doesn’t include some kind of data plan also, at least for AT&T, and an iPhone.
You have to buy a data plan. Now one thing you can do is buy the cheapest one, and “turn off cellular data usage” (that’s an option in the iPhone). Unfortunately, it can be easily turned back on so not so useful as a parental control.
Alternatively you could tell your children “no using the phone for data outside the home” and check the data usage online on a regular basis. If they use it you can see that they did.