Vinatieri first. Why?
Without Vinatieri's kicks in the Tuck Game and then in XXXVI/XXXVIII, the Brady/Belichick Pats would have been great and certainly picked up a few rings, but I think one can reasonably argue, given the randomness of the game, that there never would have been a Dynasty that went to nine SB's in 20 seasons and won six of them. Along with the Lombardi Packers, overlapping the beginning of the SB era (and maybe Paul Brown's Browns in a galaxy far far away), the BB/TB Pats are the greatest dynasty in the history of the NFL...and, I would argue, the greatest of them all.
People cite Jules' loyalty and, yes, it's a real consideration. But the Pats were never going to pay #4 what the Colts paid and...as a K...it meant something to him that they played in a dome, not only their home games but many of their Division games. No domes and a lot of crappy weather in Vinatieri's AFCE. So, I was as disappointed as anyone when he left Foxboro, but I understood and it hardly counts against him. He did what made sense for him.
Edelman has the advantage of "recency bias," especially among those on this Board who don't remember 2001. I don't think he has any bigger fan out here than me. He also had a unique chemistry with TB12. He would have been very good wherever he played, but I think Brady helped a lot to get him into the conversation of being "a great." And, of course, no one can take away "THE CATCH" in LI (as distinguished from Dwight Clark's "the Catch," for those old enough to remember both).
Finally, if we're really honest about it, Adam Vinatieri will have a Gold Jacket and a bust in Canton one day (only the haters in the media kept it from him this year). Edelman, for all his great contributions, won't get there.
I hope the idiots who make this decision call it a "tie" and let them both in at the same time.
PS: Logan Mankins was great. He belongs in the Pats' HOF too, but he's going to be up against bigger and bigger names in the next few years. Unfair.