A hole punch would splinter the board.
Tried it with GreenLee, the center hole is not the issue usually, although it depends on the outer diameter, the problem is a GreenLee does not punch in the same plane, it pierces in one corner and removes at an angle, progressively .I have used a shear to cut circuit boards (both a sheet metal shear and a shear for floor tile) without any issues, so a punch (like a Greelee punch) would probably be ok....except for the hole in the middle.
I want to make 1" round circuit boards with FR-4 (.8mm) and need a recommendation for the best way to cut them out of a bigger board. A thin wall hole saw or a maybe a punch?
The best way is to submit the design to jlcpcb.com. They are running a $2 special right now. With shipping it is under $10 to the US for 5 boards 100 x 100 or less. I think they do round at no extra charge.