Population and vegetation models often use nonlocal forms of dispersal to describe the spread of individuals and plants. When these long-range effects are modeled by spatially extended convolution kernels, the mathematical analysis of solutions can be simplified by posing the relevant equations on unbounded domains. However, in order to numerically validate these results, these same equations then need to be restricted to bounded sets. Thus, it becomes important to understand what effects, if any, do the different boundary constraints have on the solution. To address this question we present a quadrature method valid for convolution kernels with finite second moments. This scheme is designed to approximate at the same time the convolution operator together with the prescribed nonlocal boundary constraints, which can be Dirichlet, Neumann, or what we refer to as free boundary constraints. We then apply this scheme to study pulse solutions of an abstract 1-d nonlocal Gray-Scott model as a case study. We consider convolution kernels with exponential and with algebraic decay. We find that, surprisingly, boundary effects can be more prominent when using exponentially decaying kernels.