Assuming you know the other site is running php (or just about any scripting language that can get the contents of a remote text file via http), write a script for your server that generates a random key (hashed timestamp maybe?) that you can store in your db for future reference, and a second script that acts as a form action to recieve a little info (like that key you just sent) and redirect to the proper page.
On the other site, create a form where it grabs the value for teh hidden input from your script using file_get_contents() or similar, and make the link to your site do some javascript and submit your form on click (make the link to nothing - #)
User loads form at other site with uniq value in it, follows link, has unique info for that particular session so you can track the customer the whole way...