Theres no infection Im aware of that causes your fingers and toes to necrose. The only way that happens from an infection is when someone is in septic shock and the vaso pressors they need to stay alive cause their small vessels to spasm shut
Why wouldnt you suspect infectious disease? Could there be local infectious disease or insect/animal bites that are common? In California we have Valley Fever which is not know outside of certain areas. Could a local infectious agent not well known be an avenue to look down? Some resistant bacteria?