Polymer reference interaction site model (RISM) calculations are performed on melts of freely-jointed, vinyl polymer chains. The local monomer structure is modeled by three independent, hard-sphere sites corresponding to the CH2 and CH backbone constituents and an arbitrary side-chain group. The six intersite radial distribution functions and partial structure factors are computed for chains of 100 backbone units, at a liquidlike packing fraction of 0.5. It is found that the side-chain groups tend to shield the backbone sites from approaching each other at short distances. The extent of shielding increases with the size of the side-chain group. On a radius of gyration length scale all the radial distribution functions exhibit a universal correlation hole. The total structure factor, proportional to the intensity of scattering, exhibits a low-angle peak not present in a polyethylene-like melt. This low-angle peak grows in intensity and shifts to lower wave vectors as the size of the side-chain moiety increases. The isothermal compressibility, equation-of-state, and cohesive energy of the liquid are found to be functions of the local monomer structure.