Hi Jim,
In a typical year you will encounter plenty of snow in late June on the HST. Even a veteran hiker might find it a challenge. Of course, so far this year does not seem typical. But it remains to see how the next few months play out. Last year in early July, several groups turned around before the Kaweah Gap because of snow/ice. I struggled through in mid July.
Another issue in the early hiking season is stream crossings. Last year in late July they were still somewhat of a challenge for me on the HST. I would be exceedingly careful with kids 9 and 11.
Altitude is also a concern for some (independent of the time of year). In recent years I recall encountering several families turning around before the Hamilton Lakes because the kids were too sick to to continue.
In early season there will be plenty of room in the bear boxes for your stuff. Later in the summer, space is not guaranteed as others will be sharing with your group.
I think your kids are very lucky to have parents who will take them on the HST. What an adventure! But meticulous planning and preparation are key.In late August 1959 my Dad, Brother (8 years old) and I (9 years old) hiked the HST so we would arrive at the summit of Mt Whitney for the 50th Anniversary of the Smithsonian hut. We got cleaned out by a bear the first night at Panther Creek. Since one of us had altitude sickness, he did not want dinner that night anyway!
Starting in June, a phone call to the Ranger at Lodgepole (Wilderness Office) might clarify how conditions are shaping up on the trail.
Best of luck.
Jim F