Hi mahlonf,
Yes, you are correct. Method supports Authorize.net as a payment gateway and Authorize.net supports First Data as card processing platform.
Unfortunately I can not say whether or not there is a card reader that will meet your client's needs as I am not aware of any instances where one has been implemented before. If you can plug in a USB card reader, open up the Receive Payment screen in Method, swipe the card and have it populate fields then it may work. You may also be able to customize how the reader works and interacts with websites. Click here to view a previous forum post on this subject.
Should I hear of any users successfully integrating a card reader I will update both this post and the previous one.
Noel