We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and is represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. To each individual is associated a random infectivity function of its infection age, which is allowed to depend upon the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, and reflects the heterogeneities in individual characteristics and social connectivity.