heat press on cardboard??

Can you just do a stencil and spraypaint it on? If your logo isn't too detailed, you could just cut it out of stencil film. If it is highly detailed or has too many pieces that would "fall out", you could get a silkscreen stencil made and just do a quickie tape-hinge to the side of a box, squeegee it, move it to the next one, and dump it in a bucket of thinner once you're done. Monochrome is fine, even big companies rarely have a full color logo printed on their boxes. If you go through a lot of boxes, you may be better off having them custom made instead of spending the time doing it yourself, though I'm a fan of doing things myself just to learn something new, even if I'm not really saving much as long as it doesn't really hurt my bottom line. If you have your heart set on full color, you could print on labels (either clear or white) and contour-cut them to make them look better. You would then have to test the durability of the label, whether or not it half peels off or the ink rubs off while being tossed around by UP-Ex.

If you heat press remember that you're adding to your electric bill and wear and tear on your equipment with each box you make. Also I'm really not sure how well that would work.