Measurements of the growth rate of structures at z < 0.1 with peculiar velocity surveys have the potential of testing the validity of general relativity on cosmic scales. In this work, we present growth-rate measurements from realistic simulated sets of type-Ia supernovae (SNe Ia) from the Zwicky Transient Facility (ZTF). We describe our simulation methodology, the light-curve fitting, and peculiar velocity estimation. Using the maximum likelihood method, we derived constraints on fσ₈ using only ZTF SN Ia peculiar velocities. We carefully tested the method and we quantified biases due to selection effects (photometric detection, spectroscopic follow-up for typing) on several independent realizations. We simulated the equivalent of 6 years of ZTF data, and considering an unbiased spectroscopically typed sample at z < 0.06, we obtained unbiased estimates of fσ₈ with an average uncertainty of 19% precision. We also investigated the information gain in applying bias correction methods. Our results validate our framework, which can be used on real ZTF data.