Sounds simple enough.....but first, how many cards are you throwing at one time?
One toss...a whole bunch of cards? Or, one card at a time?
If you're tossing a whole bunch on one shot........look at a smaller version of a clay pigeon tosser! One spring loaded arm, locked into place with a 4-40 wire. Activate the servo....it pulls the wire, and fires all the cards.
If trying to launch them one at a time....you'll need a card tray, with a spring on the bottom, to keep the cards feeding up to rotating rubber wheel!
I'd like to see the finished product....get some video!!!!