I understand what you mean, similar to a pen plotter, where the paper is fed back and forth except its not paper, its timber and the the rollers drive the timber back and forth.
As murphy says, it would be sloppy and totally unworkable. If you were to cut parts out of a sheet, you ar expecting them to move back and forth in harmony with the sheet. I just cant see how that could happen. What about doing 3d work. What sort of roller is going to ride on the top of the rough terain youve cut? You couldnt just have the material just driven by bottom rollers, it would need a top clamping rollers too.
I am all for re inventing the wheel, as you will see from some of my previous posts, but this wheel will offer no advantages and lots of potential problems.
Please post any other ideas you have because I really believe there are thousands more ideas for this world to have. Good effort on the thinking though.
EDIT: I just noticed you mentioned a portable sign machine. Well yes they do have what you are suggesting. They cut out vinyl letters. But the cuts are only partially through and it is on a flexable material.
Being outside the square !!!