Having just done this myself, whilst I'm aware this question is from a while ago I thought it would be helpful to post my approach.
The method I have used is to execute a function in a separate file, and call this from catalog/controller/checkout/success.php
There are two methods to do this, either via directly updating the file or by using vQmod. I'm not going to go into much detail on vQmod here as it is covered extensively elsewhere (https://code.google.com/p/vqmod/). The latter is my preference as it requires no modification of the original file, however I will be covering both.
1) To modify the file directly, simply open it and on line 7 you will see
if (isset($this->session->data['order_id'])) {
immediately after this line you can add your hook. I used something like
include("catalog/controller/checkout/dropship_order.php");
sendOrderToDropship($this->session->data);
2) Alternatively, use the following vQmod code to perform the same action on the fly
<file name="catalog/controller/checkout/success.php" error="log">
<operation error="skip">
<search position="after"><![CDATA[if (isset($this->session->data['order_id'])) {]]></search>
<add><![CDATA[
include("catalog/controller/checkout/dropship_order.php");
sendOrderToDropship($this->session->data);
]]></add>
</operation>
</file>
Both of these methods will result in the calling of your sendOrderToDropship
function when the success page is loaded; and inside that function you can perform whatever action you require. By sending in the data
array, you also gain access to the order information (critically the order id) so you can perform any processing required.
Hope that helps!