Since your overall project needs to include at least a web application (for the customer order form), the simplest solution is to make this just a web application with one part for the order form and another part for the service counter to review orders. Printing from a web application to a printer connected to the web server is easy in ASP.Net, so there's no need to introduce a WinForms component.
Edit: if the web server is hosted, you could still make this an all-web app solution, but printing client-side from a browser is a pain and relatively unreliable. It's probably easier to have a web app for the customer ordering page, along with a web service that returns orders to be printed, and then a WinForms app (running on one PC, presumably) that displays the orders and prints them to a connected printer. The WinForms app could poll the web service every 30 seconds or so (you can also set up a web service so that it makes "callbacks" to a registered client "push"-style, but this is sort of a bother) and get outstanding orders that need to be printed and displayed.