Nightmare travel agent and credit cards

I have a travel agent who is causing me a lot of grief. I am not going to go into the background of the problem but I want to ask on the legality of the following scenario:

Travel agent comes with tickets to my office and hands them over. I give them a credit card but only on the condition that they do not take a copy of my card. They are only allowed to swipe the card and I refuse to let them take a copy. Is this legal? Are they allowed to insist on taking a copy or should they have to accept the credit card alone.

What are you trying to acheive here? If you want to avoid additional charges, I think they can get the number right off their copy of the receipt.

You may want to talk to your bank about this, though. They really helped me out when I had some travel agent troubles.