Please watch the posts in the General Discussion forum. Inyo N.F. may not be issuing the required hiking permits by then, and the gate on Whitney Portal Road may still be locked. Currently staying in any campgrounds is strictly prohibited. We are sure hoping it will open things up by the time you go, but that may not be the case.
If you do go, I would definitely take crampons, though you may not actually use them. Mid-June is too early to say the route is completely clear.