I believe all the ski areas in our area have seasonal programs. I work at Crotched and they have a great program there. If you're coming from Boston(ish) area it's about 1.5 hours (just under an hour from Nashua NH). I'm sure Wachussetts has one but (I've never skied there) have been told it is a bit congested. Crotched is a bit of a gem in that fashion as it is off the beaten path. No lines and a high speed quad and great grooming. It definitely has a strong race program and that is usually the only complaint we hear ("There are racers on every trail!"). That isn't actually true but they are all over the place.
I'm sure Sunapee has one and might be easier to get to as it will be mostly highway miles.
For a seasonal or even race program, you will have to figure in travel time as kids that young don't much care for getting up that early.
Plenty of folks here can answer your questions. You might want to go to the resort guide section and look at a couple places to determine what will work for you logistically and then see if you can get more info from that ambassador. Without a doubt, I'm biased towards Crotched
The biggest challenge for a parent with a young skier is keeping it fun. I've been walking that line with my daughter who is now a HS racer. For her, the social aspect is the most important part and she doesn't really care that much about the race itself. So what. She's outside and doing something healthy all winter long. Last season she logged about 5 days a week on skis. Way better than 5 days being a "vidiot".