I visited the show when I was 11 (OK, my aunt lied about my age!) and during a break got to see the set. Here's what I remember:
The selections for category (who/what/where) and amount were entered on what looked like cash register keypads--vertical columns of mechanical buttons that would lock down when pressed. If I recall correctly, there were no zero buttons--just vertical columns of buttons with 9 at the top and 1 at the bottom. Who/What/Where were on one side labeled with Dymo labels.
Years later (~1990) I got to rummage through some "spare parts" bins at Stiegelbauer scene shop in Brooklyn (
http://sai-nyc.com/) and came across some very familiar looking matrix switch sets. (BTW, Mike Stiegelbauer worked at NBC for many years). Here's what I can surmise: The number and category buttons were released by 24 volt relays. The individual wager displays (IEE one-plane readouts) were all turned on by a set of relays.
Watching the clip, I think I hear the contestants pushing the mechanical buttons for the wagers and then all of the relays engaging to turn on the displays when Art says "Declare!"
As for the magical movement of the first wager, I heard no "thunks" when the "25" moved so it may have been entered on a separate controller run by the stagehands.