Issue with querying

As I said, your code seems ok, but I think you dont get the current user beforehand.
Something like this works:

$user = wp_get_current_user(); // get current user, to get the ID

$posts = get_posts(array(
    'numberposts' => -1,
    'post_type' => 'shipments',
    'meta_key' => 'customer_user_id',
    'meta_value' => $user->ID 
));

if( $posts ){
    echo '<ul class="shipment-posts-list">';

    foreach( $posts as $post ){

        // get post data
        $post_id = $post->ID; //current post ID
        $post_title = $post->post_title; // current post title
        $post_thumbnail = get_the_post_thumbnail( $post_id ); // current post thumbnail

        // get post metadata
        $shipment_tracking = get_post_meta($post_id, 'tracking_number', true);
        $customer_user_id = get_post_meta($post_id, 'customer_user_id', true);

        echo '<li class="shipment-'.$post_id.'">';

            // example:

            echo '<h3>'.$post_title.'</h4>'; // show post title

            echo $post_thumbnail; // show post thumbnail

            echo $customer_user_id; // show customer user id

        echo '</li>';

    }

    echo '</ul>';
}

Also before you edited your question, you tried to get meta data in the wrong way.

Get meta data like:

$shipment_tracking = get_post_meta($post_id, 'tracking_number', true);

Notice the $post_id part. (instead of just $post like you had).