A fully many-body solvent influence functional for hard-sphere polymer chains in hard-sphere solvents is derived from scaled particle theory. Monte Carlo simulation of single chains in this "solvation potential" is used to calculate conformational properties of 3mers, 4mers and 10mers dissolved in 1mers. Comparison is made to the exact results of Monte Carlo simulation of the whole polymer+solvent fluid, and with the predictions of pairwise-additive integral-equation-based solvation potentials. The greater accuracy of the many-body potential suggests that at least in these systems solvent influence on polymer conformation is a strongly many-body phenomenon.