I read an old thread using a capacitor to light an LED for a few seconds, this might work for my setup. I just need a way to only illuminate the button that was pressed first. Any thoughts?

I have connected the button to a makey makey which works with a scratch script to show the fastest contestant on a screen, but I don't think Scratch can output control that I can use to switch a relay for the 12v LEDs in the buttons. Correct me if I'm wrong!

 

Use a microcontroller.

Might be able to do it with SCR's and FET's for switches.

 

Welcome to AAC!
Googling 'gameshow buzzer circuit' or 'quiz buzzer circuit' will give you lots of circuits to make one. This is a wheel which has been re-invented many times.

 

Here's my favorite circuit for that (discussed here).
It's simple, using only a small SCR and LED for each contestant, with three resistors and a Zener diode to complete the circuit.
As the simulation shows, only the first PB pressed activates that SCR and LED, which locks out all the rest.
The circuit can be expanded indefinitely by just adding a PB switch, SCR, and LED for each.

 

There are quite a few circuits that will detect the first switch closure, I designed one using relays a bunch of years ago. Exactly the same methodworks for detecting win, place, ans show for a race track. On a commercial level, there are first alarm circuits for industrial use.

 

My favorite circuit uses CD4013 flip flops and a few AND gates. It’s described in my post from almost nine years ago.

https://forum.allaboutcircuits.com/threads/game-circuit.82641/