It doesn't matter if the pizza place has a computer, as you're trying to send them a fax. I'm assuming you have some method of accessing the server, at the very least you must be able to upload files to the server.. at least that's how hosting normally works..
Okay, if your goal is to have the pizza joint receive a fax, then somehow you need to have the webserver talk to a phone line in some way. Since you're not going to be hosting the site on a machine you have physical access to, setting that up yourself isn't really an option. If you aren't hosting the site, contact customer service at the company that is. Explain to them in general terms what your trying to do, and ask if they have an 'email to fax' service, or something like it.
I will tell you, the quickest and cheapest possible solution, IMHO, would be to find a cheap computer that you can setup at the pizza place. Use it to replace their existing fax machine. Their fax machine is already using a phone line, they could get a cheap switcher so that they can be on a cheap dialup, and still get faxed orders.
This would solve so many of your problems, you wouldn't need to convert the order to a fax in any way. Also most likely going to be the cheapest way to do it, as email to fax usually costs a bit of money.
bottom line, talk to the hosters cust. service peoples, and consider a cheap box onsite at the pizza joint.
best of luck