The protections of the Bill of Rights (and the associated prohibitions on sub-federal governments) were extended to the citizens of the several states via the 14th Amendment passed after the Civil War. It says in part (emphasis added):
No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
The courts have held that "liberty" means at minimum the rights enumerated in the first ten amendments to the constitution. So a local school board, which is enforcing state laws on education, is also restricted from promulgating a policy "respecting an establishment of religion," which is understood to be any policy that advances religion. (Unless the advancement is incidental to a valid secular purpose. So for instance if "Intelligent Design" really was an important scientific discipline -- rather than a popular and political movement -- it could be included in science curricula even if it advanced religion, since teaching science in a science class is obviously a valid secular purpose.)
I think that teaching about ID in a philosophy class or a history of science class would have a valid secular purpose.
But teaching that ID is more than a conjecture in science is a fraud. At any given moment there are thousands of conjectures floating around in the sciences -- tens of thousands if you count those like ID that have no basis in evidence.