Другое действие по повторному клику на кнопке

var buttonSearch = document.querySelector(".site-nav__button-search"), 
    fieldSearch = document.querySelector(".site-nav__search-field"), 
    flag; 
buttonSearch.addEventListener("click", function(evt) { 
  !flag && evt.preventDefault(); 
  fieldSearch.classList[!flag ? 'add' : 'remove']("site-nav__search-field--show"); 
  flag = !flag; 
});

.site-nav__search { 
  display: flex; 
} 
 
.site-nav__search-field { 
  display: none; 
} 
 
.site-nav__search-field--show { 
  display: block; 
}

<form class="site-nav__search" action="search.php" method="post"> 
  <input class="site-nav__search-field" name="search" type="text"> 
  <button class="site-nav__button-search">Поиск</button> 
</form>