Some of these contact printers turn the exposure light on when the top is closed and latched. They may also have a bag that inflates to press down on the paper and neg. If it has a timer, you would set the timer, and then when you close it, it will expose for the time set.
Azo was a contact paper, and the new Lodima paper is a contact paper, but the first large production run isn't going to be available for a few months. There may be some boxes from the first test run available from www.michaelandpaula.com.
There are a few RC contact speed papers out there, but they're not easy to obtain in the US.
Meanwhile, you may be able to use enlarging papers with a contact printer like this, but your times will be very short, or you may need to reduce the exposure with ND filters above the bulb, or even just sheets of paper might work, as long as they are not close enough to the paper to add a texture to the image.