At the bottom of the membership registration form for the CoS, there is a line asking about a sponsor. What is the function of having a sponsor, and, since the CoS doesn't recruit, would I have to ask someone to be my sponsor if I wanted one?

That line is provided for those who are already in touch with a CoS member, in cases where that member is willing to provide an informal "reference." It is not necessary to have a sponsor, and seeking out a sponsor defeats the purpose.