Can a plugin add to header/footer/body content?

Hooks are going to be your best friend here.

You can filter post content by using the the_content filter for example:

add_filter('the_content', 'wse_174099_append_to_content');

function wse_174099_append_to_content( $content ) {
    //get your data
    $custom_items = get_option( 'option_name' );

    $content .= wpautop( $custom_items );

   //always return when using a filter
    return $content;
}

You can hook into the footer of the site by hooking into the wp_footer action:

add_action('wp_footer', 'your_function');
function your_function() {
    $custom_items = get_option( 'option_name' );
    echo $custom_items;
}

You could also look at the get_header and get_footer actions. The only issue is that you can’t control where the theme calls the header and footer files in – it may be a bad spot to output your code.

Hope this helps!