Okay, so it's not really a printing press.
I explained to the kids how a printing press worked in the good old days (like Gutenburg through the industrial revolution).
Then we got out abc noodles that I bought for $.80 at the grocery store (in the Mexican food section for some reason!).
We glued them onto paper backward and in reverse, and let the glue dry. Then we inked them with stamp pads and pressed a clean fresh paper down on the inked letters. The noodles are not all the same thickness, so you have to press down pretty hard--hard enough that the paper gets a sort of reverse-embossed look to it. But the letters printed out nicely.
You could print cards this way. Or make letter stamps by glueing the letters onto blocks. The hard part is that you have to press the paper onto the letters (instead of the letters on to the paper) that you want to print on. If the kids were older, I'd set them to carving new letters out of wood blocks, and we'd make a printing frame. But they're too little.