The big ski resorts have summer activities and of course lodging choices, condos and hotels, so start there..Stratton, Killington, Stowe, etc. There is a wonderful hands on science museum for kids, Montshire, on the Connecticut River,in Norwich, across from Hanover,NH where Dartmouth is located.
Burlington has a great waterfront on Lake Champlain, science museum too, near Stowe, Ben and Jerry's ice cream factory.

The Montshire is wonderful. We took our GDs ages 6 and 8 last summer. There's a section for outdoor water play so bring bathing suits. Also good idea to bring a picnic lunch.

If you stay in day trip distance of NH's White Mountains, you can add places like The Flume, Polar Caves, and Lost River.

There are some "swimming holes" around the state. One we discovered is just outside Bristol (central western side of the state) on the route that goes over the mountain to Rt 100. I think it's Lincoln Pass. It's a small river with a small waterfall people can walk under. It's cold, river water but such fun. We picnicked on a shady ledge and watched older kids and adults jump into a deep pool. Just a bit downstream was a very shallow area where you could walk across the river but the bottom is very rocky so water shoes/old sneakers are recommended. Here in NH we have a local swimming hole on a class A river with a big rock to jump off. No lifeguards. It can be a surprise to people who are only used to pools.

If you stay in the Burlington area you can also take the ferry across Lake Champlain to Ausable Chasm. Kind of pricey but they have a boat ride at the end. The walk thru a gorge at The Flume in NH is also pricey. You can also find free places to hike.

Big ski places have a lot of things to do. If you do not mind an extra drive then Jay Peak is awesome for kids. It has its own brand new indoor waterpark (pump house waterpark). They have tram rides, it is not far from burlington, and if you want to you can go to Motreal in about 2 hours. Kingdom State Forest is awesome with tons of small lakes and good hiking trails for kids. How long are you guys going for?

A good base might be in the Woodstock area, just about in the middle of the state. From here you can easily visit Billings Farm Museum, Ben and Jerry's, the Mount Ascutney Auto Road (spectacular drive and views, great picnic spots), Rock of Ages Granite Quarry, alpine slide at Pico. Don't worry - you'll see plenty of mountains. There are a few motels just outside of Woodstock, and if you want to go more upscale there's always the Queechee Inn at Marshland Farms, and the Woodstock Inn. Another idea might be a b&b farm stay. We did this many years ago with our kids, and it was one of our most memorable vacations - in fact it was our inspiration to move up here!

Shelburne Farms south of Burlington is another good place to take kids. Some short hikes with beautiful views of the lake and activities with animals. Shelburne museum is nearby but not the same place and has quite a lot to interest kids as well as adults.

Near Waterbury there is the ever-popular Ben and jerry's factory and the Cold Hollow Cider Mill. Also a nice state park for swimming.

When our children were that age (about 16 years ago) we often went to Stowe. We walked or biked the Stove Rec Path, lots of places for the kids to play and explore.

In town, south of Main Street, near the school, there is a large playground. One of us would watch the kids, while the other had a chance to shop and take a break from the children.

You can either drive up Mt. Mansfield or take the ski lift and do some hiking on the mountain.

Little River Trails area has some easy hikes. Moss Glenn Falls is an easy hike as well.

Ben and Jerry's is not far.

Mark