Could you give us some specifics on the the model of electric clay thrower you plan to use, and how it is currently actuated? At a base level, there will most likely be a low voltage trigger that actuates a relay to fire the clay. As accessible as Arduino and similar microprocessors are, I would keep it simple and use an off the shelf remote circuit switch, like this:
https://www.amazon.ca/Anntem-Wireless-Transmitter-Universal-Controller/dp/B07JG1PMGF/ref=sr_1_4?keywords=remote+circuit+switch&qid=1551218485&s=gateway&sr=8-4
If you specifically want a foot pedal, you'll need to replace the button on that particular solution with wires to a foot pedal, they can be found online as well.