I did some web searches and could not find any examples of a Libertarian school. In fact, there were no examples I could find of any political-oriented schools in the United States. There only seemed to be religious schools.
There is a strong tendency towards homeschooling coops among committed Libertarians.
The Acton Academy network of schools has concepts that are attractive to many Libertarians. The same applies to the Sudbury Valley school concept. Libertarian families are often also attracted to Reggio Emilia and Montessori schools, depending on the local community.
Closest to libertarian you can get is the Mises University by Mises Institute. in Auburn, Alabama.
There aren't many (or really any I could find which were libertarian, as opposed to conservative with free-market economics department like Hillsdale or Grove City).
Institute for Humane Studies is not a fully fledged school in and out of itself, but is a libertarian educational non-profit associated with George Mason University.
While not strictly speaking "libertarian" in a political sense, University of Chicago Economic department is of course the source of Chicago School of Economics, one of the most notable sources of neoliberal economic outlook in USA (including such notables as Milton Friedman)