You can simply hack cheap RC car. In it You will find Your receiver, and transmitter comes as part of the RC Car toy being hacked. Then You can disconnect both motors of the hacked RC Toy and replace one of them by a relay which terminals are connected to Go-Cart main circuit, so when You press/release the button on the remote, main Go-Cart circuit gets broken/activated and Go-Cart stops/starts.
If this project is somehow related to safety (say parent looking after a child in a Go-Cart), a Dead Man Switch can be implemented. When button on a remote is released or Go-Cart goes too far, relay breaks the circuit and cart stops.