The Antarctic continental shelf (ACS) hosts processes that impact the climate system globally, which has motivated ongoing efforts to characterize its state, circulation and variability. However, the nature and consequences of eddies over the ACS, and their contributions to the budgets of heat and freshwater, remain systematically understudied. This study uses hydrographic measurements collected from instrumented seals, supported by a high-resolution model of the southern Weddell Sea, to characterize eddies and their role in vertical heat transport around the entire ACS. A key finding is that eddies are ubiquitous, and exhibit frequent (10-50% of hydrographic casts) occurrences of O(1) Richardson numbers, indicative of submesoscale variability. However, along-track density power spectra exhibit wavenumber dependences of k^-3, consistent with quasigeostrophic turbulence. Approximately 1% of the points in the surface mixed layer satisfy conditions favorable for symmetric instability, though its prevalence is likely higher than this due to the relatively coarse resolution of the seal tracks. Vertical heat transports, estimated from a regional model-calibrated parameterization of submesoscale restratification, are largest in dense shelf regions, which have previously been identified as key sites of warm water intrusions onto the ACS. These regions also exhibit the largest seasonal cycles, with elevated winter eddy activity and heat fluxes accompanying the formation of high salinity shelf waters. These findings indicate that eddies may contribute substantially to ACS heat and tracer budgets, and motivate further study of their role in determining the pathways and fate of heat that intrudes onto the ACS.