Update custom category fields front-end

the function wp_insert_term returns the newly created term id (or WP_Error on error), so once your create your term you need to store it’s ID and then you can save the “extra fields” using get_option, update_option something like:

if($_SERVER['REQUEST_METHOD'] == "POST")  {
    $clientName = $_POST["clientName"];
    $term_id = $_POST["clientName"];
    $clientKeywords = $_POST["clientKeywords"];
    $clientSlug = $_POST["clientSlug"];
    $parent="";

    $errorType="Category "". $clientName .'" Added!';

    $term_id = wp_insert_term($clientName, 'category', array(
        'description'=>$clientKeywords,
        'slug'=>sanitize_title($clientSlug),
        'parent'=>$parent
    )); 
    if (!is_wp_error($term_id)){
        //save here
        $tag_extra_fields = get_option(MY_CATEGORY_FIELDS);
        $tag_extra_fields[$term_id]['field'] = strip_tags($_POST['field']);
        $tag_extra_fields[$term_id]['field2'] = strip_tags($_POST['field2']);
        update_option(MY_CATEGORY_FIELDS, $tag_extra_fields);
    }
}