Kindergarten is still considered a school under the Federal definition of a school zone, so someone with a firearm and no concealed carry permit could be tried in Federal court.
The Federal school zone (anywhere within 1,000 feet of a K-12 school) does not apply to a person with a state-issued permit or license to carry said firearm. You still can't go onto school grounds, but can be within 1,000 feet.
It would be good for the laws to match up, though.
In VA, you can only go onto school grounds if you remain in your vehicle and the firearm remains concealed. So for drop-off and pickup of children. And of course, only if you have your Concealed Handgun Permit.

What is the point of rules that are not enforced?