I can imagine two ways to accomplish this. The first is lighter on your side in terms of bandwidth, but requires more access to the client's site. The second requires very little access to the customer's site, but proxies all data through your site, potentially increasing your bandwidth costs:
1) At a minimum, you would need to install a .htaccess file, and some variety of PHP, CGI or other program to do the page retrieval on myclientssite.com. Myclientssite.com would also need to enable mod_rewrite to accomplish this. Requests are routed to the PHP script, which checks for the existence of a matching file from your server. If it exists, it essentially acts as a proxy, and forwards the data. (Your server should engineer the correct data, this client script will just pass that along) If your server gives a 404, the script could either serve the local page itself, or via a redirect rule, hand processing back to apache.
Just a word of warning: This kind of processing can get expensive after a short period of time, so you should include some sort of cache list, or file listing from your server so the script can tell after a short time what files it needs to fetch, and which it should pass through.
2) Another solution might be to have your client create a DNS alias to your site (protected.myclientsite.com) and have your server reverse-proxy their data. The client would then set an ACL to only allow your site to read files from the actual protected site. This will increase your bandwidth usage however, and so is less than ideal if the protected data is say... large pictures, video files, etc.