In this article, a special class of distributed stochastic nonconvex optimization problem is investigated. Each agent in the network only has access to a local stochastic weakly convex objective function and can only communicate with its neighbours in a random time-delay environment. For solving the considered problem with the distributed communication under time-delay interference, the distributed weakly convex delay-tolerance algorithm (DWDTA) with diminishing step-size and fixed step-size are proposed, respectively. Specifically, we show the convergence of the DWDTA with diminishing step-size by using Moreau Envelope measurement and demonstrate the linear convergence of the DWDTA with fixed step-size under sharpness condition. Our convergence results explicitly characterize the influences of the weakly convex function and the random time-delay interference on the convergence performances, respectively. Finally, numerical results are worked out to verify the effectiveness of the DWDTA.