Do any of the royals belong to a book club.?

The royals are not known for being well-read.

Possibly the Duchess of Cornwall. One of her pet causes is literacy, and she promotes it rather enthusiastically. She's said to be a great reader, though I doubt she reads the sorts of new age-y books her husband's into.

Princess Michael of Kent writes books and takes part in lecture tours/clubs.

http://www.princessmichael.org.uk/

Prince Charles has written books, but if he belongs to a book club, it would be a private one, only for people he invites. He probably just discusses things with Camilla and friends informally, no need for a club.

Camilla is a literary advocate and has headed meetings where children read and discuss books.

No, they get people to read for them. Mostly Ladybird books, peoples friend type mags, and lots of Hunting, shooting and fishing nasties.