If the POST contains the first_name and last_name fields, add to the customer created through WooCommerce.

my_woocommerce_new_customer_data.php

<?php
/**
 * If the POST contains the first_name and last_name fields, add to the customer created through WooCommerce.
 */
function my_woocommerce_new_customer_data( $new_customer_data ) {
	if ( empty( $new_customer_data['first_name'] ) && isset( $_POST['first_name'] ) ) {
		$new_customer_data['first_name'] = sanitize_text_field( $_POST['first_name'] );
	}
	if ( empty( $new_customer_data['last_name'] ) && isset( $_POST['last_name'] ) ) {
		$new_customer_data['last_name'] = sanitize_text_field( $_POST['last_name'] );
	}
	return $new_customer_data;
}
add_filter( 'woocommerce_new_customer_data', 'my_woocommerce_new_customer_data' );