Over the next three months, Moorgate will consolidate all three of its branches in Luton, Chelmsford and Stratford into the one central facility.
Managing director Neville Simpson said the move would not involve cuts to Moorgate's 55 staff.
Moorgate's current HQ in Stratford falls within the proposed 2012 Olympic site and the firm said it decided to make its move before an enforced relocation took place.
More than 300 businesses will have to relocate out of the Olympic zone, but Moorgate said it had "made the most out of the situation".
"The new site is bang in the middle of all of our business," Simpson said. "Having our stock and staff all under one roof will have great advantages for both our costs and efficiency."
Simpson said that once the move is complete, Moorgate would be looking for further acquisitions outside of London, investing in a new fleet of lorries and upgrading its IT systems.
Moorgate announces relocation plans
Moorgate Paper has announced a 3m investment in a new 11,000sqm facility in Enfield, north London.