I was able to figure it out myself, here is the code:
<div id="fullpage">
<?php
//for each category, show 5 posts
$cat_args=array(
'orderby' => 'name',
'order' => 'desc'
);
$categories=get_categories($cat_args);
foreach($categories as $category) {
$args=array(
'showposts' => 5,
'category__in' => array($category->term_id),
);
$posts=get_posts($args);
if ($posts) {
echo '<section class="vertical-scrolling" data-anchor="' . $category->name.'">';
foreach($posts as $post) {
setup_postdata($post); ?>
<div <?php if(has_term( 'slide', 'fp_type' )): ?>class="horizontal-scrolling"<?php endif; ?>data-anchor="<?php the_title(); ?>">
<article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
// Here comes the content //
</article>
</div>
<?php
} // endforeach;
echo '</section>';
} // endif;
} // endforeach;
?>
</div>
It seems to work quite fine.
@Michael´s answer helped me a lot