I'm no sports fan, so anyone more knowledgeable, feel free to correct me if I mess up the details here. (Like you'd hesitate ...)
In American football, a team has four "downs" in which to make a required amount of progress (10 yards). If they cannot progress that far in their four tries, the other team gets the ball and they can then attempt to move the ball and eventually score.
If it becomes apparent to a team that they are not going to get far enough and the other team will get the ball, the first team wants this to happen as far from the other team's goal as possible. One way to do this is to "punt", which is to drop-kick the ball downfield (away from the other team's goal).
So in football, a punt is what you do when you have concluded that success is impossible and you want to minimize the effects of failure. Thus it has entered general speech to say thigs like, "This project is a failure. Let's just punt and see if we can re-use some of the work on another project." (It's also common to expand it a little to "fall back and punt".)
Or in this case, "We don't want to use a noun, so let's punt and use a pronoun instead." They're saying that a pronoun is the "next best thing" to using a "real" noun. Of course it's not really inferior to use a pronoun in the sense that it's a mistake or you would be better to use a noun, but they're giving the idea that a pronoun is a substitute for a noun.