Yes, the NFL should investigate Sean Peyton's actions following the Saints game with Williams and the Rams. If there is any question about impropriety or anything illegal or unsportsmanlike, then there should be consequences. If nothing is uncovered, then everyone can move forward. But there is nothing wrong with doing the investigation.
It is my belief that Sean Peyton should be investigated for his actions, due mostly to the questionable way in which he has conducted himself in the past. After the recent scandal from several years ago, I do not think anyone should trust Peyton anymore without doing some independent investigation first.
The NFL has a very poor track record for investigating and punishing itself. Investigations like the one into Sean Peyton and the Saints will only lead to speculation and accusation, but will not lead to any findings of wrong-doing. Except in the case of gross and obvious misconduct, the NFL should let its coaches do their job.
The NFL should not investigate Sean Payton's actions after the Saints game with Greg Williams and the LA Rams. There was not anything fraudulent or potentially criminal about what he did, and it does not have any repercussions on the sport. NFL investigative resources could be better used elsewhere to look into players and coaches that are actually involved in suspicious activity.