The White Mountains during the Summer is quite nice, but dry, high, cold and windy. No bears or other creatures to worry about, no wag bags. No snow during the Summer.
For children with no hiking experience, there are much nicer and easier hikes all along the eastern side of the Sierra. Plenty of water and beautiful scenery. We take our five year old grand-daughter hiking with us a lot, with no worries. No wag bags except at Mt. Whitney.
There are bears, but they are not likely to be a problem for you, as long as you don't leave food out overnight around your campsite. Even at that, they are much more interested in your food than in you.
There are other animals, like marmots, squirrels, pika, birds, etc. which your kids will enjoy seeing.
Here is the local Inyo National Forest website,
https://www.fs.usda.gov/inyo, which has most of the information you at likely to need. Give them a call; they are very helpful.
