Smooth Poincaré operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We introduce analogous constructions of discrete Poincaré operators acting on complexes of cochains and Whitney forms. Naturally these operators allow one to show the vanishing of cohomology spaces for these complexes. We particularly focus on having the operators be constructive (i.e. computable) under various assumptions on the underlying domain or simplicial complex. We showcase discrete Poincaré operators based on singular simplices and realized constructively using Whitney forms on domains which are star-shaped with respect to a point and more generally on contractible domains. We also showcase combinatorial versions of the discrete Poincaré operators on simplicial complexes which are collapsible and also those which are "discretely contractible." Finally, we show how the discrete Poincaré operator on star-shaped domains can be modified to obtain a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC). This is based on joint work with Johnny Guzmán, Anil Hirani, and Bingyan Liu.