Understanding and enhancing thermal transport in polymers is of great importance, and is necessary to enable next-generation flexible electronics, heat exchangers, and energy storage devices. Over the past several decades, significant enhancement of the thermal conductivity of polymeric materials has been achieved, but several key questions related to the effects of molecular structure on thermal transport still remain. By studying a series of electrospun vinyl polymer nanofibers, we investigate the relationship between thermal conductivity and both molecular chain length and side group composition. For polyethylene nanofibers with different molecular weights, the measured thermal conductivity increases monotonically with molecular chain length, as energy transport along molecular chains is more efficient than between chains. The observed trend is also consistent with structural characterization by Raman spectroscopy, which shows enhanced crystallinity as molecular weight increases. Further, by comparing the measured thermal conductivity of vinyl polymer nanofibers with different side groups, we found that phonons travel along polymer chains more effectively when the side groups are either lighter or more symmetric. These experimental results help reveal the underlying correlation between the molecular structure and thermal conductivity of polymer nanofibers, providing valuable insights into the design of polymeric materials with enhanced thermal conductivity.