Can't comment because I am newbie.
To add/comment @reeds and @securityOranges answers:
This seems like it could be easily done with switches as semi low tech option.
Make circuit such as:
Battery to switch to switch to switch to led back to battery.
Then one can even build and demonstrate its fair working in front of all participants.
Wires could be as pretty much as long as needed.
Lights/leds could even be added to next to each switch.
Thou I would probably just give people cardboard box to hold their hand while being in same room.
Buttons can be used to ensure that even if system is physically stolen during votes they return to their original state fast enough that no information can be gained.
Then just people to look at the clock and everyone will vote for ~10 seconds when clock hits certain time.
Edit:
I build demonstration of this:
https://i.sstatic.net/pihWA.jpg
In brief:
Above two of three buttons are pressed but no light.
Bellow all of the three buttons are pressed and thus light came.
And here hopefully picture to clear connections between components:
1, 2 and 3: From battery to first button
4, 5 and 6: Buttons
7, 8, 9 and 10: connection to resistor
11: the resistor
12: the LED
13:Connection back to "Battery"
I used Arduino as my battery but could have been any other method to deliver power for the led.