For every credit card purchase, the merchant needs your credit card number/expiration date and one of these three items:
(1) Your name and address, including the zip code and sometimes your county of residence (AVS, or Address Verification Service).
(2) Your credit card's CVV number.**
(3) An imprint of your card, or an electronic swipe of your card.
Now, if I recall correctly, online stores such as Powell's Books (powells.com) uses method (1). Porn sites usually use method (2), or uh, so I've heard. Places like Pizza Hut and Gamestop use method (3).
For those merchants using method (3), if they can't swipe your card, then they need to make an imprint of your card. Having a delivery driver doing it is no different than having a clerk in front of a register doing it. They're still transferring the embossed portions of your credit card to a piece of paper. There's also no difference between doing a thermal rub of your credit card's information and using a manual credit card machine (other than needing to buy a machine that does the same thing as rubbing a pen across thermal paper).*** And the merchant would need an imprint of the whole front side of your card. It's to prevent mistaken or fraudulent chargebacks.
Here's the relevant information from Visa's website:
For key-entered or voice-authorized transactions, make an imprint of the front of the card. The imprint proves the card was present at the point of sale and protects your business from potential chargebacks if the transaction turns out to be fraudulent. The imprint can be made either on the sales receipt generated by the terminal or on a separate manual sales receipt form signed by the customer.****
Since you don't use your CVV number to make most pizza delivery purchases online or over the phone, the imprint is required. This is because the merchant had to manually enter your 16-digit account number and expiration (or the website only asked for the 16-digit number and expiration, and not including the CCV code on the back of your card).
So if you've bought some pizza with your credit card over the phone or through the internet, please understand that if the delivery person actually follows procedure and asks for an imprint, she or he is just doing their job.
*Except for you few people that keep yelling out your pin number to the cashier when you hand them a debit card.
**Usually a 3-4 digit number in the signature box of the card. Different companies may use another name for this number.
***Personally, I prefer the thermal paper, since that will fade away in a few months, and imprinting machines use ink which doesn't fade away.
****Page 21, from this PDF file.