The hike up to June Lake S of Lee Vining is nice and totally doable by kids.
I checked and June Lake has a road to it. What trail you are refering too that is really good? I have an account with trails.com which has all kinds of trails described in that area.
Sorry - didn't say that very well. The area around June lake is nice, with hikes, etc. Here's a description - click the figure links to see a map:
