Sorting WP Auctions by date/price?

<a href="https://wordpress.stackexchange.com/questions/77197/<?php bloginfo("template_url'); ?>/auction_price.php">Order by price</a>

auction-price.php is inside the template directory.

Then you can’t include wp-blog-header.php using require(‘./wp-blog-header.php’) inside template directory.

Therefore do_shortcode() is not working without WordPress.

The proper way to use ajax in WordPress is via admin-ajax.php

Read More at Codex:

http://codex.wordpress.org/AJAX_in_Plugins

For example, in your functions.php:

function wpse_77197_auction_price(){
    echo do_shortcode('[wpa_list_widget mode="list" order="price"]');
    die();
}

add_action('wp_ajax_price_action', 'wpse_77197_auction_price');
add_action('wp_ajax_nopriv_price_action', 'wpse_77197_auction_price');

In your template file:

<div id="tabs">
  <ul>
    <li><a href="#tabs-1">Order by date</a></li>
    <li><a href="<?php echo admin_url('admin-ajax.php'); ?>?action=price_action">Order by price</a></li>
  </ul>
  <div id="tabs-1">
    <?php echo do_shortcode('[wpa_list_widget mode="list" order="date"]'); ?>
  </div>
</div>

Update:

Try to handle the ajax request yourself in the “wp” or “init” action as explained in https://wordpress.stackexchange.com/a/53491/19683

add_action( 'init', function() { 
  wpse_77197_register_shortcode_ajax( 'wpse_77197_auction_price', 'price_action' ); 
} );

function wpse_77197_register_shortcode_ajax( $callable, $action ) {

  if ( empty( $_POST['action'] ) || $_POST['action'] != $action )
    return;

  call_user_func( $callable );
}

function wpse_77197_auction_price(){
    echo do_shortcode('[wpa_list_widget mode="list" order="price"]');
    die();
}