Custom Field Order by Last Name and First Name

Your code has a few issues, some unrelated, but generally, you need named meta queries to order by multiple meta keys:

$posts = get_posts( [
    'post_type'      => 'soldier',
    'posts_per_page' => 5,
    'paged'          => $soldier_paged,
    'meta_query'     => [
        'last_name_clause'  => [
            'key'     => 'last_name',
            'compare' => 'EXISTS'
        ],
        'first_name_clause' => [
            'key'     => 'first_name',
            'compare' => 'EXISTS'
        ],
    ],

    'orderby' => [
        'last_name_clause'  => 'ASC',
        'first_name_clause' => 'ASC',
    ],
] );