An integral equation approach to understanding connectedness percolation in macromolecular fluids is extended to investigate semiflexible linear homopolymers. A coarse-grained threadlike approximation is developed for persistent chains. Results are presented for the percolation threshold, and number of intermolecular contacts at the threshold, as functions of the polymer stiffness. A substantial reduction of the percolation threshold is observed, and a dependence of this threshold on the contour length similar to that for rigid rodlike particles is found, even for large ratios of the contour to the persistence lengths provided the latter exceeds the range over which neighboring monomers are viewed as belonging to the same physical cluster. Accounting for chain stiffness leads to pair correlation functions which, at small separations, display features akin to those observed in fluids of rodlike particles, resulting in much lower critical volume fractions than for fully flexible molecules.