Your armpit has lots of lymph nodes which can swell and become tender to indicate infection. In any case, whatever is happening to his arm isn't a normal reaction to a shot. He should call his doctor or head over to the ER. Hope he feels better!
Armpit swelling in men's can be caused by internal disease cancer, infection lymph swelling due to disease immediate attention by a physician is required
Armpit swelling in men, might be from allergies from food. Or your not comfortable with your dress or you have allergies you get from deodorants ^^