Add a search box in a WordPress menu that will be rendered through the native wp_nav_menu() function.

wp_nav_search_box.php

// Add a search box in the main navigation menu
function nav_search_box( $items, $args ) {
  if( $args->theme_location == 'main-nav' ) {
		$items .= '<li class="nav-search">
			<form role="search" method="get" id="searchform" action="' . home_url( '/' ) . '" >
				<input class="nav-search-field" type="search" value="' . get_search_query() . '" name="s" id="s" placeholder="Αναζήτηση" />
			</form>
		</li>';

		return $items;
	}
}
add_filter('wp_nav_menu_items','nav_search_box', 10, 2);