AutoZone's statement that it's either an engine *or* transmission problem is a very general statement because many issues affect both. A problem in one system can trigger a fault in the other. For example, a failing torque converter (transmission) can strain the engine, leading to codes in both systems.
The fact the manager said it's a code that can't be read directly is important. A standard OBD-II scanner, the kind AutoZone uses, primarily reads engine codes. Transmission codes *might* be read by this scanner, but often require a more specialized tool that connects directly to the transmission control module (TCM). The location of this connector varies greatly between makes and models of vehicles.
To get this diagnosed properly:
* Find a shop with a specialized scan tool: You need a mechanic with a scanner capable of reading transmission codes. This is especially true if the vehicle has an automatic transmission. A simple OBD-II scanner is unlikely to fully diagnose the problem.
* Describe the symptoms: Don't just tell the mechanic "the light is on." Explain how the car is driving: Is it slipping? Does it hesitate? Are there unusual noises? This information will help pinpoint the problem.
In short, the light itself isn't enough information. You need a proper diagnosis using the appropriate tools. Ignoring it could lead to further, more expensive damage.