The AD's office is much bigger than most of us think. There are hundreds of people who work for Fulmer. Some of them support the football program 100% of the time, and a lot more work football along with the other sports.
So anyone in Marketing could come up with an idea like this video clip of CJ. Or anyone in the Football Operations division. Or the Media Relations folks. Or anyone, really. They would pitch the idea to one of the leadership in the AD office, and eventually have to get the approval of a few folks including probably Todd Watson (Director of Football Operations). Then turn it over to the Media Relations branch to turn the idea into an actual video, a product, and then publish it.
Bottom line is, the AD's office is a business center. With a host of intermediate leaders and coordinators we never hear about. And they're the ones who get all this stuff done. Generally without ever having to bother Pruitt or Fulmer. It's all below their noise threshold.
So the short answer is, the AD's office. If you have to put a name to it, Todd Watson is probably as close to an "owner" of that twitter account as anyone.
They do a good job.
University of Tennessee Athletics