If you're looking at early in Carter's presidency, and if you're looking at the Mafia, then you might want to look at Paul Castellano and Aniello Delacroce. In 1976, Carlo Gambino passed on the mantle of heading up his organisation to Castellano, which annoyed Gambino's No 2, Delacroce. Delacroce accepted the decision, for all of a year or so, and things went downhill.
The details of how it went downhill are not particularly important, but Castellano had gained the backing of both the Westies (an Irish-American mob outfit) and the Cherry Hill Gambinos (a Sicilian American ditto); these were notorious groups, noted for being "wild cowboys", even among the various branches of the Casa Nostra.
Now Castellano was only one capo, but that grouping does have the potential to get out of hand. I don't know if it's what you want, but these guys were not especially concerned about collateral damage. On one occasion, there was a shoot-out in a crowded diner involving them and another group, in which 500-1000 shots were fired. Astonishingly, no-one was injured in this (I am not sure how this happened; by the laws of chance, that many bullets in that small a space with that many people must have resulted in someone getting hit. Apparently not, so score one for the stormtrooper school of marksmanship). Assuming a turf dispute gets out of hand, with that degree of marksmanship, and if Jimmy Carter happens to be in the vicinity, and his Close Protection is a bit sleepy (it happens. Close Protection is boring), then you could end up with Carter dead as a result of collateral damage, which would somehow feel appropriate for him.
You'd need a reason for Carter to be in New York, which shouldn't be difficult. You'd need a dispute between Castellano and another Capo, which is almost a given. You'd need a location that both could conveniently be at; the Giants Stadium to watch a game might be strangely appropriate.
However, there are a few problems. Hoffa had been imprisoned in 1967. His imprisonment might make his becoming Vice President tricky. His being pardoned and on good terms with Nixon, and - in as much as political parties bothered him - owed a huge debt to the Republicans, might make it difficult for a Democrat who'd promised to clean out the corruption left by Nixon to appoint him. But these pale into insignificance with the problem posed by the fact that he disappeared in July 1975, and was undoubtedly killed then. Being both missing and dead are generally significant obstacles to becoming Vice President.
This is incredibly interesting actually, and I think it would work well with the story i'm writing. As far as Hoffa goes, in this TL he doesn't follow quite the same career path, and doesn't wind up in prison (although he does face a similar demise as IOTL). Thank you for the information on the mob though, it's very useful!