The MGB probably has one of the best classic car spare parts availability you can find. Just about every spare part is available new even complete bodyshells. Used parts are in abundance at the many autojumbles around the UK every year especially the big annual events.
The parts suppliers I have used quite regularly are listed below.
MGB Hive a well established supplier which I regularly use and can recommend
MGOC Spares another well established supplier which I have used many occasions – just wish they would give some discount to their members
Anglo Parts that managed to supply me a very rare gearbox part recently
Peter Edney MG Cars has a well stocked online parts store as well as many other services
Midland Sports and Classics have provided good parts to me
Leacy has a vast amount of stock
Moss a major supplier in all parts MG and others
Rimmer another major supplier
Brown and Gammon also a large supplier of parts