To do something vociferously is to do it vehemently and loudly. People vociferously support things they believe in.
To be vociferous is to be strongly and vocally in support of something. To do something vociferously is to act that way in some situation. During football games, coaches vociferously pester the referees, and fans vociferously support their teams. In politics, supporters campaign vociferously for their candidates. Martin Luther King vociferously worked for civil rights. This is a word involving strong verbal support for something. You can’t act vociferously if you’re silent.