You could meet and not divulge anyones contact details, but for her at least I imagine meeting you would be a start.
But how would that work in practice?
It would be wrong to share info about the cousins parents and siblings if the parents and siblings werent on board with the connection being made...
.... but then doesnt that sort of make the meeting defunct? If the whole purpose of this stranger wanting to meet is to find out about their family from you?
Last weekend, in Rutland, the first statue in Britain of the late Elizabeth II was unveiled.