We present the first results of an ongoing program we are undertaking with the Wide Field Camera of the Hubble Space Telescope (HST) to understand the physical origin of the enhanced star formation seen in moderate-redshift (z approximately 0.3-0.4) cluster galaxies. Deep HST exposures have been obtained for the central regions of two rich compact '' Butcher-Oemler '' clusters, AC 114 at z = 0.31 and Abell 370 at z = 0.37. Both of these clusters have been subject to extensive ground-based spectroscopic and multiband imaging studies with a significant fraction of confirmed members being seen in the active or ensuing phases of a starburst. We have used the HST images in conjunction with the ground-based data to examine the morphology of individual cluster members, in particular these '' starburst '' objects. We find that those blue members that display spectral evidence of active or recently completed star formation are predominantly disk-dominated systems whose abundance is greater than that seen in the cores of present-day rich clusters. Furthermore, approximately 55% of the galaxies in this category in both clusters show convincing evidence of dynamical interactions, in contrast to a 20%-30% rate of occurrence among the red population. While similar conclusions had been drawn from high-quality ground-based data, we demonstrate the unique role HST can play in identifying interacting galaxy pairs which have separations less than the current resolution limits of the best ground-based images. There is convincing evidence in these first two clusters we have examined that interactions and mergers play a major role in inciting the star formation activity associated with the Butcher-Oemler effect. Of equal significance is the morphological nature of the numerous red members in our HST data set which show various spectroscopic and photometric indications of previous starburst activity, including strong Balmer-line absorption. Most of these galaxies appear to be undisturbed and isolated with a normal E morphology. There is no convincing evidence that these are merger products. Although larger samples are still required, we conjecture that the Butcher-Oemler effect may involve at least two physical processes arising from the hierarchical growth of clusters: galaxy-galaxy interactions and environmentally induced star formation arising from the hierarchical merging of clusters. Much work remains to be done to understand exactly how the fraction of disk galaxies seen in distant rich clusters declines to its present low value.