Use a filter on menu items that have children

There is indeed a filter, walker_nav_menu_start_el will be able to handle it for you!

Take this code for example:

function wpse356896_filter_primary_nav_menu_dropdown( $item_output, $item, $depth, $args ) {

    // Only for our primary menu location.
    if ( empty( $args->theme_location ) || 'primary-menu' !== $args->theme_location ) {
        return $item_output;
    }

    // Add the dropdown for items that have children.
    if ( ! empty( $item->classes ) && in_array( 'menu-item-has-children', $item->classes ) ) {
        return $item_output . '<span class="dropdown"><img src="https://wordpress.stackexchange.com/questions/356896/chevron.svg"></span>';
    }

    return $item_output;
}

add_filter( 'walker_nav_menu_start_el', 'wpse356896_filter_primary_nav_menu_dropdown', 10, 4 );