We experimentally investigate the periodic vortex shedding dynamics in a highly oblate Bose-Einstein condensate using a moving penetrable Gaussian obstacle. The shedding frequency f (v) is measured as a function of the obstacle velocity v and characterized by a linear relationship of f (v) = a(v - v (c)) with v (c) being the critical velocity. The proportionality constant a is linearly decreased with a decrease in the obstacle strength, whereas v (c) approaches the speed of sound. When the obstacle size increases, both a and v (c) are decreased. We discuss a possible association of a with the Strouhal number in the context of universal shedding dynamics of a superfluid. The critical vortex shedding is further investigated for an oscillating obstacle and found to be consistent with the measured f (v). When the obstacle's maximum velocity exceeds v (c) but its oscillation amplitude is not large enough to create a vortex dipole, we observe that vortices are generated in the low-density boundary region of the trapped condensate, which is attributed to the phonon emission from the oscillating obstacle.