In US English should I say
You have already activated more sites than your license allows
Or
You have already activated more sites than your license permits
In US English should I say
You have already activated more sites than your license allows
Or
You have already activated more sites than your license permits
In some degree, 'permits' sounds a wee bit more formal. Given the formality of the sentence, I would suggest 'permits'. But then again, you were specific about AmE, and a little digging showed that the use of 'allows' is more widespread in American literature.
This Google Ngram on 'license allows' vs 'license permits' clearly shows that the usage of 'allows' is far more extensive.
Note: The comparison has been done over books written in AmE.