Characterizing the mechanisms and galaxy properties conducive to the escape of ionizing (LyC) emission is necessary to accurately model the Epoch of Reionization and identify the sources that powered it. Using Hubble Space Telescope data, the Ly alpha and Continuum Origins Survey (LaCOS) is the first program to obtain uniform, multiwavelength subkiloparsec imaging for a large sample (42) of galaxies observed in LyC and enable statistically robust studies between LyC and resolved galaxy properties. Here, we characterize the morphology and galaxy merger properties of LaCOS galaxies and investigate their connection with the escape fraction of LyC emission fescLyC . We find strong anticorrelations between fescLyC and size (r20, r50, and r80) measured in filters containing emission from star-forming regions, and with the asymmetry and clumpiness in F150LP, a filter tracing UV continuum and Ly alpha. We find that >= 48% of LaCOS galaxies, and >= 41% of LaCOS LyC-emitters are visually classified as galaxy mergers. Galaxies robustly identified as mergers in LaCOS are at advanced stages of interaction, close to coalescence. The fescLyC properties of robust mergers and low-probability mergers cannot be differentiated statistically, and we only find significant difference between the two populations in terms of their sizes and LyC luminosity: robust mergers having larger values. We conclude that (i) fescLyC tends to be larger in galaxies with a small number of compact, centrally located, UV-emitting star-forming regions, (ii) mergers at advanced stages of interaction represent a sizable fraction of LyC-emitting samples at z similar to 0.3, and (iii) mergers can facilitate the escape of LyC photons from galaxies.