Nothing in the Code about it. I would only worry about physical damage from it being hit by someone digging. So if it were me I would strive for 18-24 inches deep.
Couple of questions:
What environment is this in? Tropical rain forest, desert, ocean-side
What type of terrain do you have? Soil, clay, rock
What community/zoning area? Residential, commercial, industrial
The only thing you will find in the code is the fact they have to be protected from physical damage. Just like any other GEC in the book. There is no minimum depth.
You are right there is nothing in code about it, the way the NEC is laid out, its a chapter 8 application, and chapter 8 requirements are mostly exempt from the rest of the NEC requirements, in this case, burial depth.
If this is a permanent installation, for that soil possibly the best constructed ground would be a concrete encased electrode (CEE) also known as a Ufer ground after its developer.
You would not need to meet the NEC criteria, just provide a concrete pad with sufficient area and conductive rebar. Although a lightning strike might damage the concrete.
This is a common reoccurring question on the Ham radio forums.
Because there are four interconnecting 'ground' systems involved.
1] Tower lightning protection ground.
2] GEC (Grounding Electrode Conductor) ground.
3] EGC (Equipment Grounding Conductor) ground.
4] Radio Frequency antenna ground. (only some antennas have a Earth ground)