I'm not certain what the definition of "cheap" is but a quick Google
search on "raster to vector software" turns up one package for $80USD
(http://www.rastervect.com/) and WinTopo Freeware which is free
(http://homepage.ntlworld.com/heatons/softsoft/wintopo/index-free.htm)
I couldn't find any software where they pay the user to use it.
The expensive packages are the commercial ones used in engineering and
architecture that are hopefully unnecessary for converting MIDI
printouts.

Indeed, if your media does not bend, you need a printer with a
"straight-through paper path".  I'm not aware of any home user printers
with that feature but an email to manufacturers like HP & Epson may
turn up one.

I always refill my inkjet cartridges.  It reduces the cost by a factor
of 5-10 to 1.  It would also allow me to experiment with printer's inks
that are designed to print on glossy cardboard stock if I needed to.
It might require a drying booth or throwing the prints in the oven.
Air drying time for these inks can be 30 minutes.

One last idea:  Get an inexpensive laser printer.  I saw one for
$100USD recently and used ones can be "cheap".  Reverse your print
image before printing, disconnect the fuser or stop the print before
it reaches the fuser by opening a panel at the "right" time.  Remove
the "print" very carefully, lay it face down on your cardboard, roll it
with a screen printer's roller, fix the transferred toner with a spray
available from art supply houses (Used for pastel drawings).

This same thing could be down with a copier or laser fax machine.
You may need to pre-treat the cardboard to accept the toner.  There
may be some spray that will do this.  Talk to a professional printer.
All things are possible.  Good luck.

Jim Divoky