The check engine light on our Subaru keeps coming on. The first time, the computer showed that it was an issue with the gas cap. They reset the light and we were fine. The second time, apparently, they could not check if it was the gas cap b/c the tank was full. So, we had them just change out the cap. Well, now it has happened a third time. Any ideas??
What needs to be known is exactly what code(s) is involved. It could be that someone is misinterpreting the code, or codes.
So they can’t check because the tank was full, huh? 
Your Subaru is having diagnostic problems not electrical issues. There is no code that means gas cap is the issue. Their however is an entire evaporative system code where the gas cap is one of numerous potential problems.
Have they checked the charcoal canister?
Also what year/mileage is this Impreza?
Right, it did say evaporative system. I mispoke.
The dealership said that the evaporative system could not be checked with the tank full.
It is a 2007 Impreza with about 52000 miles on it.
I do not know the exact code.
I do not think that the charcoal cannister has been checked.