Archives of the Royal Economic Society institutional papers from its genesis are kept at the London School of Economics & Political Science Library Archive. More information is available about their holdings via their website http://www.lse.ac.uk/library/archive/ following the links for the Royal Economic Society and you can apply to them directly. Permission to access the RES archive is embodied in the hand list provided by the LSE which tells you which files are open and which closed. The handlist can be accessed here.
A collection of papers by John Maynard Keynes is kept at King's College Cambridge which has an online catalogue that lists Keynes' RES-related papers.
Information on past members may be available from the archive of membership records of the Society which are currently waiting to be catalogued. Please contact the RES office for more details on email@example.com.