Affect can mean a number of things relating to internal feelings; mood, demeanor, "caring". If you say someone has limited affect, that means they don't display much emotion, and perhaps don't feel a lot about external events.

I've not heard of affective vulnerability specifically before, but the most obvious meaning, as has been already offered here by a few people, is that it means

"an unusual sensitivity to other's emotional states or to events that could make you emotional"

(although IMO, that basically describes being human)