The way many clubs work is that members draw up a shortlist, each month, either by each nominating a title, or one person nominating several, and they vote on which one they'll read, ready to discuss at the next meeting - where they'll take a vote on the next title, and so on.
The library service I work for operates a book group service, whereby groups are given a list of all the titles available to them and one person in the group is responsible for ordering mutiple copies of books from the library, collecting them and returning them afterwards. Obviously this requires a little more forward planning on their part, especially with very popular titles, but it is a well-used service, so it must be getting something right.