The Constitution, for the most part, does not restrain the individual; it restrains the government. The first amendment does not generally apply to private parties. Exceptions exist
On my
personal property, I can restrain any number of First Amendment rights, such as not allowing guests to practice their religion or not allowing them to watch adult videos.
The states, as well as their proxies in local government,
are restrained through the Supreme Court's incorporation of
most of the Bill of Rights via the Fourteenth Amendment. Amendments not held to be incorporated against the states are the Second, the Third, part of the Fifth, the Seventh, and part of the Eighth. An analysis of the First Amendment's incorporation can be found here:
http://1stam.umn.edu/main/historic/Incorporation%20Chart.htm
Students are not subject to the same First Amendment protections as ordinary adult citizens. However, their right to access non-obscene, non-pornographic material
is constitutionally protected. The LGBT filter employed by the Tennessee schools is restraining that right.
Schools and libraries are required under CIPA (Children's Internet Protection Act) to use web filtering software as a condition of receiving certain federal funds. The Supreme Court has upheld (wrongly, in my opinion) that this condition is a valid exercise of the spending power of Congress.
I don't believe
any censorware can perfectly balance students' First Amendment protections with the requirements of CIPA. The filtering software used in Tennessee goes above and beyond the requirements of CIPA by blocking access to non-obscene, non-pornographic information. Interestingly, the software allows access to anti-LGBT information, such as "reparative therapy" and "ex-gay" ministries. This makes me suspicious that the filtering is targeting pro-LGBT sites, which in turn makes me believe that it's not simply a software deficiency.