Using the Facebook Javascript API you can execute a friend request

Facebook Friend Request via js

        FB.ui(
	        { 
		        method: 'friends.add', 
		        id: fbid // assuming you set this variable previously...
	        }, 
	        function(param){

		        console.log(param);

                // If they cancel params will show: 
                //    {action:false, ...}
                // and if they send the friend request it'll have:
                //    {action:true, ...}
                // and if they closed the pop-up window then:
                //    param is undefined
	        }
        );

/*
The callback script can then simply performs an ajax call to your server where 
you save info about the action, if needed.

You can test this by using the javascript console app on Facebook:

http://developers.facebook.com/tools/console

Paste in the script above, including the <script> tags, or click the "Examples"
button on the bottom of the text area and find the "fb.ui — friends.add" example.
*/