This link is pretty good
NY Pistol Permit information by County - AR15.COMto get an understanding of how the different counties issue.... the guys try to keep it as up to date as pos. A lot of counties are backed up right now so I would add about 3-4 months onto any time that is listed!
I just saw you said her family is in Buffalo, I seached this out because I was going to move up there. From what I have read, Niagara County (on the border of buffalo) issues unrestricted as apposed to erie county (where buffalo is)