In case anyone has a similar need in the future, I'll add a tool. The question doesn't indicate the circle size, but if you're using a plastic stencil, I'll assume the circles aren't very big.
You can get inexpensive sets of hollow punches sold as gasket punches. They're typically a collection resembling this:
Image courtesy Amazon
I bought a cheap set years ago from Harbor Freight that goes up to 1-1/2" diameter (they no longer show it, but hardware stores, Amazon, and other places have similar sets).
They can punch out a circle, or align an outer and inner cut to create a gasket or washer. You put the material on a surface like scrap wood, position the punch, and hammer it until it's all the way through.
They will punch any soft material, and cut all the way through with a clean edge, so there isn't any tearing. I've punched washers out of a roll of weatherproofing felt.
BTW, for this kind of activity, you can also punch out other kinds of shapes. You can buy collections of shape cutters like this:

Image courtesy Amazon
They're often sold as leather crafting tools, but they work with fabric and other soft materials. Note that some of these are on the small size, intended for creating holes and slots rather than pieces.