Relevanssi AJAX search

relevanssi-ajax.php

<?php

add_action('wp_ajax_relevanssi_search', 'rlv_ajax_search');
add_action('wp_ajax_nopriv_relevanssi_search', 'rlv_ajax_search');
function rlv_ajax_search() {
  $query = new WP_Query();
  $query->set('s', $_POST['s']);
/* Any other parameters you want to set, for example:
  $query->set('post_types', 'page');
  $query->set('numberposts', 100);
  */
  $posts = relevanssi_do_query($query);
  echo json_encode($posts);
  wp_die();
}

/* the jQuery code */
var search_term = $("#s").value(); // or wherever you get the search term

var data = {
  'action': 'relevanssi_search',
  's': search_term
}
jQuery.post(ajaxurl, data, function(response) {
  results = JSON.parse(response);
};
      
/*
That's about it, this is how I would approach this.
Remember that ajaxurl is not defined on the front-end by default:
  https://codex.wordpress.org/AJAX_in_Plugins#Ajax_on_the_Viewer-Facing_Side
*/