In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method to multiscale photon transport. In this method, the photon free streaming and scattering processes are treated in an un-splitting way. The photon distribution is described by both discrete simulation particle and analytic distribution function. By accurately recovering the multiscale modeling of the unified gas-kinetic scheme (UGKS), the UGKWP method presents a smooth transition for photon transport from optically thin to optically thick regimes according to the cell's Knudsen number. In the optically thin regime, the UGKWP method performs as a Monte Carlo type particle tracking method, while in the optically thick regime it recovers a diffusion process without particles. The proportion of wave-described and particle-described photons is automatically adapted according to the numerical resolution and local photon scattering physics, i.e., the so-called cell Knudsen number. Compared to the discrete ordinates-based UGKS, the UGKWP method requires less memory and does not suffer from ray effect. Compared to the implicit Monte Carlo (IMC) method, the statistical noise of the UGKWP method is greatly reduced, and the computational efficiency is significantly improved in the optically thick regime. Several numerical examples covering all transport regimes from the optically thin to optically thick ones are computed to validate the accuracy and efficiency of the UGKWP method. In comparison with the UGKS and IMC method, the UGKWP method may have a several-order-of-magnitude reduction in computational cost and memory requirement in solving some multiscale transport problems.