I've lately seen different events, like Fort Worth Stock show and NAPE Expo, where schools like Tech and TCU have won out at these events. I haven't seen anywhere if A&M sent a team or even where to look for that info.

Does anyone have any information?

Like this one?

TAMU car at 1:45,
Accepting an award at 4:05

We also have a Clay Target team that competes under the ACUI organization, and TAMU Trap and Skeet is the reigning national champion!

Civil engineering has Steel Bridge, Concrete Canoe, GeoWall, and water turbine competitions, where you compete against other schools. You can also enter research papers in virtually any field. Most of the engineering majors have multiple ways to compete.

As far as academic and trivia-type competitions, those are usually staffed by schools with larger and more significant liberal arts programs. The competitions A&M participates in are often more focused at the lucrative game of identifying potential employees for companies. General trivia competitions don't generate as many sponsorship dollars, so A&M doesn't seem to pursue them as much.