We present galaxy circular velocity functions (GCVFs) for 34 low-redshift (z<~0.15) clusters identified in the Sloan Digital Sky Survey (SDSS, Cat. <J/AJ/123/567>), for 15 clusters drawn from dark matter simulations of hierarchical structure growth in a {LAMBDA}CDM cosmology, and for ~22000 SDSS field galaxies. We find that the simulations successfully reproduce the shape, amplitude and scatter in the observed distribution of cluster galaxy circular velocities. The power-law slope of the observed cluster GCVF is ~-2.4, independent of cluster velocity dispersion. The average slope of the simulated GCVFs is somewhat steeper, although formally consistent given the errors. We find that the effects of baryons on galaxy rotation curves is to flatten the simulated cluster GCVF into better agreement with observations. The cumulative GCVFs of the simulated clusters are very similar across a wide range of cluster masses, provided individual subhalo circular velocities are scaled by the circular velocities of the parent cluster. The scatter is consistent with that measured in the cumulative, scaled observed cluster GCVF. Finally, the observed field GCVF deviates significantly from a power law, being flatter than the cluster GCVF at circular velocities less than 200km/s.