Add keywords meta to index.php page

I guess you use metadata to store keywords for other pages.
One way if to use some page for the source of homepage keywords.

Other is to add theme option for home page keywords:

https://blog.templatetoaster.com/wordpress-settings-api-creating-theme-options/

Afterwards:

add_action('wp_head', 'my_wp_head'));
function my_wp_head() {
    if ( is_front_page() ) {
        $keywords = get_option('homepage_keywords');
        echo '<meta name="keywords" value="' . esc_attr($keywords) . '">';
    }
}