The issue of whether the U.S. Vice President should salute is somewhat nuanced and tied to military tradition and protocol. Generally, saluting is a custom reserved for military personnel as a sign of respect and recognition of rank. In this context, since the Vice President of the United States is a civilian, one might wonder why or if they need to salute.
It is customary for the President and the Vice President to be saluted by military personnel when they are in uniform because of their positions as Commander in Chief and senior leaders of the country. However, the Vice President does not typically initiate salutes since they do not hold a military rank. Instead, it is a sign of respect from military personnel towards the office the Vice President holds.
Ultimately, the Vice President does not have a responsibility to salute others but is usually acknowledged with salutes from military personnel as they represent the government and its authority.