Its literal translation must be "executing body" or "implementation body", but I am afraid that there is no appropriate equivalent English. Therefore, it must be better not to translate it literally. For example, if you translate "そのイベントの実施主体は市です", you should say, "The event is organised by the municipality", "The municipality organises the event", or "The municipality is responsible for the event", rather than saying, "The executing body for the event is the municipality".