I can’t enqueue Suggest.js in frontend – not added

Provided you’ve confirmed that WP’s native ‘suggest’ script is actually being rendered on the page properly (ie. you’re using the correct template for home vs front-page), sometimes to avoid script conflicts on the page, you need to wrap your script in an anonymous function. So something like:

<script>
(function($) {
    $("#my_input").suggest("<?php echo get_bloginfo('wpurl'); ?>/wp-admin/admin-ajax.php?action=ajax-tag-search&tax=my_taxonomy", {
        multiple: true,
        multipleSep: ","
    });
})(jQuery)
</script>

Ideally, this should also be separated off into its own .js file and registered/enqueued properly on the wp_enqueue_scripts hook with ‘suggest’ declared as a dependency:

wp_enqueue_script( 'my-suggest', 'path/to/my-suggest.js', array( 'suggest' ), false, true );