EGCJFlyer:
Clubs: Big one at Seathing - Waveney Flying Group, you can find them on the internet. Had a bit of a reputation of being a bit "exclusive", perhaps because they have a waiting list and expect time of their members. You will probably have to serve time sweaping the runway before you are allowed to fly!
Otherwise, south of Dereham is Shipdham, where I believe there is a club but don't know about hiring. Further out Further out is the McCauley Flying Group at Little Snoring with a PA28 (I think). Beccles - yes I know its Suffolk! - also hires aircraft, I believe. It is fairly accessible from Norwich.
Old Buckenham is a nice field near Attelborough. They have a number of groups there. I know there is a PA28 Archer/180 there though don't know if they have shares available. I don't think that the flying school rent out aircraft, but give them a ring.
You can also try the flying schools at Norwich Airport, such as Anglian Air Centre, who certainly hire out aircraft: Warrior and C150, though these are also used for training, so can't say what availability is.
If you are after a group, there are at least two PA28 groups at Norwich, one of which is mine! If you have tailwheel time I can put you in touch with the Chipmunk group which flies from Felthorpe a few miles NW of Norwich, or a Cub group from Priory Farm, in South Norfolk.
PM me if you need to know more. May also know of aircraft for sale if that is the direction you want to go.