It is likely in this case that to "punt for" means to "punt on behalf of". To "punt" refers to the football metaphor for giving up, for reverting to plan B, etc.
As the leading answer has detailed, "punting" in American football is the act of kicking the ball downfield to the other team, voluntarily giving up possession of the ball in the face of an impending failure to convert to a first down. This is not a forfeiture of the game, but rather a calculated trade-off done in hopes that the team might score on some future possession. To give up possession is not ideal - ideally, one's team would advance downfield and either convert to a first down, or score by way of a touchdown or field goal - but under certain circumstances, such as on fourth down, it is better to give up possession and place the ball as far away from one's own goal line as possible than it is to attempt and fail to convert to a first down, which would still yield possession to the other team, but would also leave them in much better scoring position.
Similarly, when presented with a problem for which the ideal solution cannot be implemented, one might "punt" and revert to a less-than-ideal solution. Example: "We would have liked to install a state-of-the-art swimming pool in the dormitories, but alas, we had to punt and buy the students passes to the local rec center." This usage connotes a deferral of the problem or a postponement of the ideal solution.
When one punts for someone or something, one is taking on a specialized substitute role with the implication that the substitution is not exact and the results will differ from what the original entity might have achieved. The implication is that the entity doing the "punting" is less than ideal, but still adequate.
In the case of pronouns "punting" for nouns, the pronouns are behaving as substitutes for nouns (much like a punter would substitute for the quarterback in a football game on fourth down), but are doing so with less precision than the nouns otherwise would. The pronouns still, however, get the job done.