GRB 010921 was the first HETE-2 gamma-ray burst (GRB) to be localized via its afterglow emission. The low redshift of the host galaxy, z=0.451, prompted us to undertake intensive multicolor observations with the Hubble Space Telescope with the goal of searching for an underlying supernova (SN) component. We do not detect any coincident SN to a limit 1.33 mag fainter than SN 1998bw at 99.7% confidence, making this one of the most sensitive searches for an underlying SN. Analysis of the afterglow data allows us to infer that the GRB was situated behind a net extinction (Milky Way and the host galaxy) of A$_V$åisebox-0.5ex 1.8 mag in the observer frame. Thus, had it not been for such heavy extinction, our data would have allowed us to probe for an underlying SN with brightness approaching those of more typical Type Ib/c SNe.