There are two different issues - immunity and indemnity. Immunity is freedom from prosecution for a crime and can't even be offered by the NFL. It is the government or prosecutor who offers you immunity from prosecution in exchange for your testimony or cooperation, usually against another. bigger criminal they want to nail. I don't see how the NFL can offer this unless they get the prosecutors in Foxboro MA (whom I assume would be the governmental agency prosecuting Matt Walsh for crimes like theft of Patriots property or recording other persons without their permission) to agree to it. If there are potential federal crimes they would have to get the US attorney to agree too. The Pats can push for prosecution at the local level if he did steal tapes and illegally record conversations. That could be very expensive for someone who now lives in Hawaii.
Indemnity is an agreement to pay for your defense costs and any damages awarded in a civil suit against you. Essentially, Matt Walsh is looking for the NFL to become his insurance company if the Patriots (or any persons connected with them, such as BB or Scott Pioli) sue him for damages for anything he says, does or turns over. For instance, if he has said that he has a tape of the Rams walkthrough (or even implied it and not corrected it), and does not and the Pats (or BB or Pioli) have been damaged by that (i.e. loss of reputation, actual economic damages, etc.) they could sue him for things like, libel, slander, negligent or intentional misrepresentation, etc., etc. It could get very expensive for Mr. Walsh, or the NFL, if the Pats decide to go that route. It would also pit Roger Goodell against Bob Kraft. Not a very good scenario for either one.
This could all be taken care of if the Pats were involved and agreed not to push for prosecution of Walsh on anything and not to sue Walsh for whatever he does. However, the Pats don't seem to be involved. They are waiting on the sidelines, it appears. If I were them, when the time was right, I would call Goodell and say if you want us to play ball you agree that no more penalties be handed out no matter what this kid has (which we still think is nothing). Whatever he has is at least six years old and didn't happen on your watch, so let it go. Let him and Specter have their moment in the sun and then we can get back to football.
Goodell doesn't seem that smart, however.