Stochastic nonconvex optimization problems with nonlinear constraints have a broad range of applications in intelligent transportation, cyber-security, and smart grids. In this paper, first, we propose an inexact-proximal accelerated gradient method to solve a nonconvex stochastic composite optimization problem where the objective is the sum of smooth and nonsmooth functions, the constraint functions are assumed to be deterministic and the solution to the proximal map of the nonsmooth part is calculated inexactly at each iteration. We demonstrate an asymptotic sublinear rate of convergence for stochastic settings using increasing sample-size considering the error in the proximal operator diminishes at an appropriate rate. Then we customize the proposed method for solving stochastic nonconvex optimization problems with nonlinear constraints and demonstrate a convergence rate guarantee. Numerical results show the effectiveness of the proposed algorithm.