I would still be nervous about any public computer. I just saw a huge thing on identify theft, and all the ways including just Google, that people can get your info. I am very careful even at home, but I would still just call in the order rather than using a public computer.
This is funny because my husband and I were just talking about this sort of topic over breakfast. He was telling me that a co-worker had his identity swiped and there was a person in Florida, we are all in California, who actually had several credit cards and a HOUSE.....all with his credit information. Crazy!