Round punches and dies are available from a number of manufacturers.
Using them will be far easier than making your own and they will
outperform anything made of oil-hardening steel.  Dies are available as
die buttons, which can be press-fitted into the die block.  In this
application, you can probably allow the die button to stand a bit proud
of the die block, so that you will be able to sharpen it several times
without having to remove stock from the larger die block.

Chose a punch the size of the desired hole.  The die will be oversized.
The difference between punch and die diameters is twice the clearance.
Clearance is an important variable.  In punching metal, 10% of stock
thickness is considered a typical clearance.  Increasing the clearance
reduces punching force.  Decreasing it reduces burrs on the underside
of the punched hole.  For cardboard, I would guess that the ideal would
be a very small clearance, since you are shearing, not the bulk
cardboard, but the individual fibers in the cardboard.  Be careful,
though.  You can only have a small clearance if your setup has very
little lateral play.  If a tool-steel punch clips a tool-steel die on
the way in, both will chip and be ruined.

Jonathan Herz