Vito Corleone When I first saw The Godfather, I thought perhaps LSI, but SEE is especially prominent if you see Part II with Vito as a younger man. He knows exactly how to conquer the hearts and minds of individuals using his aptitude in Fi (doing people 'favours' when they do him one) coupled with penchant for expansiveness Se. A brilliant problem-solver, using people's feelings as a solution; he knows how to appeal to the masses. Clearly Fe-able, but I think he has Se ego, and prefers Fi (consider his feelings about the Five families in Part I after Sonny's death ("This war ends now").
Michael Corleone Despite his being one of the two protagonists in both Part I and Part II I actually find him quite difficult to type. I'd like to say LSI (IJ temperament and strong Se seem quite evident), but I get the impression that he couldn't care less for Fe (consider especially his introduction at the beginning of the first film at his sister's wedding. He doesn't get involved, he just sits there). Perhaps, just perhaps, a Fe PoLR type would be likely. Maybe ILI. This could be a longshot though. There is the poor relationship with Kay, which seems to suggest a pure lack and devaluing of Fi.
Sonny SLE is pretty evident. He has a clear interest in dominating his environment, and is focused on the here and now protection of his family. Expansive and willful; he's completely upfront. A Se subtype for sure.
Tom Hagen Seems to operate well with Michael; only occasionally do they argue or have a problem. Perhaps activation partners. He could be an ESI if Michael is an ILI, or an IEI if Michael is an LSI. I see a lack of Se, but no problems with it; he could easily value it.
Fredo Essentially quite passive, but very dependent. SEI could be a possibility, although he could value Se (he seems particularly inept in it). Wants to be respected and seen as someone powerful, but clearly lacks this image. Perhaps IEI would be more likely taking into consideration his possible Se valuing nature.
Kay Doesn't care for the Mafia's strict Ti rules; probably an EII. Michael has scant regard for her feelings, and she reacts badly to this. Loyal, despite the many occurrences, but walked all over. Perhaps ESI.
Connie More minor than the other characters. ESE is a possibility.
Hyman Roth Maybe an LIE. Very business-minded; seems not concerned with a Fe atmosphere.
Add your own interpretations and/or other character typings.