I recently tried a booking on the DL website for a friend's child where the boy's grandfather was paying for the ticket and received the same response from the website.
I went through the website, having created a new skymiles account and was hoping for an online booking bonus miles scenario.
In the old days, the third party would pay for the ticket as a PTA (pre-paid ticket advice). A service charge was always applied for this "service."
I believe this is a nice little revenue generator for the airline and DL is not keen on losing these extra $$'s. I also believe that their policy reduces the possibility of fraud, although many other carriers, their websites and the websites of others (ie, Travelocity) will allow this type of payment.