Stopping a gokart with your feet is dangerous, just dont do that.
It destroys shoes
You could break an ankle or leg, or blow out a knee or worse.
It looks like some sort of drum brake. You could repair it and continue using it, but the drum (what I could make out) also looks badly worn and ribbed, so I'd update and upgrade.
Disc isn't really an option unless you want to do some fabricating, welding, and switch the wheels out as well... there's not even a guarantee you can find a drum to match your hubs.