Darn, I just left Vashon after two weeks!
We like KVI beach at Tramp Harbor. At low tide there is a great sandbar, then an incredibly rocky area that is often hidden by the tide. The boulders house some amazing sea stars. As you come down to the beach and cross the bridge, there is also a creek with lots of sealife to explore, it's usually full of kids and moms.
Ober park is fun, always moms and kids there too. The library is next door if you feel like going in for a story.
Point Robinson is fun, too, the lighthouse has tours on Sundays.
Check out the Saturday market, and the concerts in the park as well. All are very family friendly.