Scioly.org

I honestly don’t know where to start with the build. I’ve considered a couple of designs, such as constructing a device similar to Roller Coaster or making an hourglass with multiple chambers. The only design that seems to make sense is a pendulum clock with an escapement that rotates a gear, and each full rotation counts as a single signal. However, that design wouldn’t be very adjustable, requiring a different gear for each possible required time. Is there a way to fix this, or might there be a better design I’m overlooking?

Yeah, reading the discord I think that the mechanical escapement is the best design. However, you may want to consider adjusting the length of the pendulum for each trial instead of adjusting the wheel. You could also try a water clock (something like a bucket that repeatedly fills with water and tips after so much time).

Can we use gears purchased online to construct the device or does it have to be created using 3-D printer/manually?

The rule you're concerned about is 3b: Commercial counters, tally devices, timepieces or their parts are not allowed.

It would be easy for an ES to rule that a commercially-manufactured gear wheel was a "part of a timepiece".

Commercial gears are fine as long as the gears did not come directly from a commercial clock. So if you buy a pack of random gears from amazon then it's fine because they have no direct relationship with a timekeeping device. However, if you take apart a grandfather clock, for example, and use the gears from it then it would be illegal. Make sure to include where you got the parts from so the ES can't argue with you.