Is it permissible to view her photo after she swore to her brother that she would not send her photo to anyone, and does she have to offer an expiation for her oath?

You must avoid sins to preserve the legitimate sisterly relationship and love for the sake of Allah. Requesting pictures of friends is not advised. It is impermissible for your friend to send her picture in violation of her brother's covenant or in breaking her oath, because sending pictures is neither a necessity nor an act of closeness to Allah. Photography is forbidden, especially photographing women, due to the temptation it entails, and a woman is awrah (private, to be concealed). Pictures published online are not safe and may be misused. Many upright women share friendships and love without seeing each other's pictures.