Yes, there are hole-punching dies for grommet machines. Specifically, they're sold as cutting dies for grommet machines, as you've mentioned, and also hole-piercing dies. They screw in to the press the same way the grommet dies do.
Unfortunately, the threading for the attachment of the die to the press is not universal, there can be differences between brands according to user reports. On top of that, ALEKO doesn't include any specs to go by (or even a product manual), making it more difficult to answer if Stimpson-compatible dies, for instance, will work with this product.
A cutting die for your size (#2 for instance), will work, in theory, on your press, but you may have to try to obtain one from different manufacturers. ALEKO themselves do not supply anything other than the 5 grommet dies #0-4.
I have asked ALEKO, on the only outlet where it seems they reply publicly to users, whether or not additional dies are compatible.