The PIN is:

3352752

The result came from a great group effort. I hope I’ve correctly identified and summarized the process followed.

From Wesley Situ:

The two wheels will be turning in opposite directions. Three wheels have the ‘ascending t notch’ (for lack of a better description) going counter-clockwise (Red, Cyan, Blue) and three have them going clockwise (Green, Magenta, Orange). We can then pair them up with one another as the ‘ascending t notch’ will cause those things on the sides to bend downward, then snap quickly back up (think of the Wheel of Fortune wheel). I’d like to think that this causes some kind of sound to play so we’ll hear a bunch of clickety clack.

With that information, we looked into pairing the cogs together. From Poolsharker, GentlePurpleRain, and LeppyR64:

There are three pairs of gears that complement each other.

Blue 21202122201000121222121
Green 01020100021000101000101

Orange 10122021210221001021222
Magenta 10100201012001221201000

Cyan 12202101002010210010200
Red 10020121220212012212022

All pairs line up on the bumps (the 1s) and no two triangles (the 2s) ever overlap. There are a couple places where the flat areas overlap. A graphical representation of the above:

With that knowledge, we looked into what it could mean. From LeppyR64:

The machine generates:

Morse Code
Loading a gear on the left side will generate a dash
Loading a gear on the right side will generate a dot

Here are the possible orientations and assignments:

Orange/Magenta
..-. . -.. -- -. ... | FEDMNS
--.- - .-- .. .- --- | QTWIAO
... .- -- ..- . .-.. | SAMUEL
--- -. .. --. - -.-- | ONIGTY

Red/Cyan
--.- . ..-. .- .. .-.. | QEFAIL
..-. - --.- -. -- -.-- | FTQNMY
..-. .. -. .-.. . -.-- | FINLEY
--.- -- .- -.-- - .-.. | QMAYTL

Green/Purple
. ... . . .-. ...- | ESEERV
- --- - - -.- ---. | (Error)
-... .-. . . ... . | BREESE
.--- -.- - - --- - | JKTTOT

Orange/Magenta – SAMUEL
Red/Cyan – FINLEY
Green/Purple – BREESE
His last name is of course, MORSE

From here, we focused our attention on the PIN pad. From Verence:

Morse invented the telegraph in 1836 and died in 1872. 1872-1836 = 36=6^2, 1872*1836 = 3436992 This could mean that the “t” stands for “telegraph” but what is “i”?

And extended from the train of thought of Verence, I found:

What if the Sun and the Cross symbolized birth and death? Morse was born in 1791 and died in 1872.
1872-1791 = 81 = 9^2. 1791*1872 = 3352752.
The PIN is 3352752.