I'm assuming that your website is hosted remotely on a webfarm someplace, whereas your printer is stored locally.
Possibly the best answer is NOT to send anything directly to your printer, but rather to send them someplace where they are (automatically) printed from. This will protect you from issues such as a printer problem causing you to lose an order.
I would suggest that you need a program running on your PC or local server which will wake up every few minutes, look for any new orders, and print any that it finds.
There are a number of ways of achieving this:
Your site could store the orders in a database (which it should anyway) and your local program could then query that database for new orders, format the info into a printout and send it to the printer.
your site could store the orders, ready for printing, in textfiles in a directory and your local program could then search that database for new orders, and send them to the printer.
your site could email the orders to a special address, and your local program could log into that mailbox, download the order info, format it and send it to the printer.
How do those ideas sound?