Who Has the Power to Make This Change?

The President can issue proclamations recognizing specific months for particular causes. For example, presidents have issued proclamations for Pride Month annually since Bill Clinton in 1999. A petition could urge the President to issuing a proclamation designating June as Veterans Month instead. 
President Donald Trump, as of April 2025, could be petitioned to issue an executive proclamation recognizing June as Veterans Month.
Petition members of Congress, such as the chairs of the Senate Committee on Veterans’ Affairs (currently Sen. Jon Tester, though subject to change post-2024 elections) or the House Committee (Rep. Mike Bost), to introduce a resolution.
Pride Month is heavily supported by corporations, media, and nonprofits. A petition could target influential organizations (e.g., veterans’ groups like the American Legion or VFW) to advocate for Veterans Month and pressure cultural shifts.
Petition the American Legion or Disabled American Veterans to launch a campaign for June as Veterans Month.
Progressive groups, media, and corporations could frame the petition as an attack on inclusivity, mobilizing counter-petitions or protests.
Feasibility: No single entity “owns” Pride Month; it’s a decentralized observance. Replacing it would require coordinated efforts across multiple levels of government and society, which is logistically daunting.