The nonlocal diffusion problem with volume constraints has been widely applied in many fields, such as fracture of composites, fracture of polycrystal, nanofiber networks, image analysis and financial engineering, the accuracy of the existing numerical methods, including the finite element method for the piecewise constant element, finite difference method, quadrature and particle methods are low. To obtain a high order numerical method, a high order finite element method is proposed for a 2D nonlocal diffusion problem with volume constraints, and the stiffness matrix of the numerical method is extracted from a new matrix $B$, which is easily computed. And the coding principle of the elements and the expression of code in the numerical calculation nodes are given. Also, the numerical example illustrates the convergent order of the new finite element method for 2D nonlocal diffusion problem with volume constraints. It is worth pointing out that it is not trivial to solve the 2D nonlocal diffusion problem with volume constraints.