What you are looking for are "flatbed cutter" or "flatbed cutter plotter".
The "flatbed" part is important.
Standard vinyl cutter do not need a flatbed since the vinyl sticks to a backing paper. This enables the cutter to roll and unroll the material(vinyl) without having to worry about loose pieces of material jamming the mechanisms. The backing paper also gives the stock a consistent stiffness. This is not the case with paper. Since it looses stiffness the more cuts you make into it and all these loose parts are not fixed to a backing, it would jam the vinyl cutter very quickly.
I have no experience with these flatbed paper cutters. So I can't recommend a particular model, but looking at how they work, I'd imagine there are limitations on thickness and weight of the paper that they can cut. They probably can't cut extremely detailed designs on thicker stock...
What I do have is experience with laser cutter. They have the advantage that you can cut nearly any thickness of paper, paper-stock or cardboard without much limitations on the design.
Lasercutters can cut paper extremely fast, so it probably won't be too expensive.
If you need to do A LOT of equal cuts you can also look into die cutting.
There is a lot more start-up costs, though, since the cutting is done with a steel-rule die in a press. The dies are similar to cookie cutters