JPL Horizons On-Line Ephemeris System provides easy access to key solar system data and flexible production of highly accurate ephemerides for solar system objects.
Osculating elements at a given epoch are always an approximation to an object's orbit (i.e. an unperturbed conic orbit or a "two-body" orbit). The real orbit (or the best approximation to such) considers perturbations by all planets, a few of the larger asteroids, a few other physical usually small forces, and requires numerical integration.
JPL ephemerides do not use things such as periods, eccentricities, etc.