Not everyone has the inclination, ability or time available to build an aircraft. The only route then is to buy a part-built aircraft for completion, a ready-flying aircraft or an aircraft in need of restoration.
For part-built LAA projects, the new owner/builder needs to continue following the LAA amateur-build process. For already-flying LAA aircraft, the new owner simply takes on the responsibilities of the old owner for looking after the airworthiness of the aircraft. The restoration of an aircraft might range from dusting off the cobwebs and giving the engine a good service, to a full strip down to component parts and fabricate missing or damaged parts.