I assume they didn't cut Lee's arm because they want him to die
If we're looking for reasons story wise...
Lee chopped off that guy's leg on ep2, although it was just to free him not because he was bitten, and he died.
Maybe he didn't want to risk dying faster from blood loss, or having 1 less capable hand/not being able to help at this point that he needs to find Clementine. But the idea probably didn't crossed his mind because of Clem.
And If he had chopped it off there would be no doctor to fix him since Vernon
is gone as we see plus Lee thinks he's the one that took her anyway.