Since it all comes as one the 16gauge wire might be fine. I usually use 10 gauge for main power to the relay, 12 gauge to each light, and use 14 gause to run the trigger wire to and from the switch. Though for $6 for everything it would be worth a try. If you had to go buy everything seperately it would cost you about $12-15, depending on where you bought the wire. I find buying the wire by the foot from a hardware store is cheaper than buying iot on a role at an automotive store.

I usually get the plain 5 pin relays from Autozone or Kragen's, they usually run about $5. You will need to get a box of female flat connectors to go with the relay.

JC Whitney also sells the complete harness.