Sorry, I was on the road so it took me a bit to get to this question. This is a question I get a lot, and to be honest about it when I've answered it in private communication it does seem like the parents don't really heed my advice, probably because it doesn't match up with what they have in their mind. So, I'll just put it out there and you can take it as you like.
There are maybe two questions: What gym would give my kid or kids Thai style padwork, or correction? I think you could get pretty good attention at most big gyms that frequently take on westerners. Giving padwork or correction to kids isn't all that different than giving it to adults. Thai padholders tend to be very patient with kids, and pretty good at simplifying technique, as most fighters were trained first as children.
The bigger question is: What gyms might train my kid/kids like a Thai boy (or girl) would be trained? And that is a very different question. Thai kids are not trained with lots of padwork or correction. And of all the gyms I've seen in Thailand I've not seen one (with fleeting exceptions), other than my own gym Petchrungruang, that lets western kids mix in with the Thai boys, just as if they were one of them. My gym seems very unique, not only in what it does, but that it's been doing it for a long time. By my advice you want your kids training with other kids. That's how Thais learn. There is no substitute.
The important factors though are that this takes time. You don't go into Thai style training for a week and come out looking like a ninja. You don't learn tricks, or moves, or even combinations. You are put in a big pack of other kids and forced to just spar and clinch to find your own level. It can be a big adjustment for western boys who might be used to being special in their home gym. But gradually, and I've seen this first hand, boys learn to start responding in more Thai ways. It's not just a question of technique, but of spirit.
My gym has had several long term western boys go through it, some of them at the gym for more than a year. Some are boys that come every year for a stretch. But all of them experience a kind of immersion teaching. If the gym you are considering does not have lots of Thai boys to play and work with then what you are probably looking at is technical training of some sort which is a very different thing.