Almost all exchanges do it. Most of the exchanges require valid ID before others don't. One of the examples of a exchange that doesn't require ID is Circle. Most of the time they are instant and you get your bitcoins immediately after verification of 3D security (if enabled or required).
You can also vist
the wiki page to get more information about how the different exchanges work and what they require.
Like Aubert said, never trade creditcard directly with a person. The information on the creditcard dosen't change, and the person would be able to do all sort of things with its information.