The First Amendment states, in part, that the government shall not abridge one's freedom of speech. The Museum of the Confederacy at Appomattox and the Virginia Museum of Fine Arts have both previously declined to fly the flag, and as a result, the VF has organized protests in response. Neither of these museums is under any obligation to provide a means for the VF to exercise its members' right to freedom of speech. However, on their newly leased property, the VF is perfectly at liberty (har har) to say whatever they like. Obviously they have to meet local zoning standards and local laws, but assuming they do, the group is free to display their flag under the First Amendment.
Edit: Special thanks to my friend Byron, who helped make this a more accurate post! "...The Second Confederate Navy Jack is exactly the same as Tennessee's Battle Standard but is a brighter colored blue and not navy blue. The Confederates definitely weren't known for their originality lol." Thanks Byron!!