Electronic structure methods enable first-principles calculations of the properties of molecules and materials. Applying numerically exact electronic structure methods to systems relevant to chemistry is computationally intractable due to the exponentially scaling cost of solving the associated Schrödinger equation. In my talk, I will describe several methods we developed to reduce this cost, specifically by incorporating Monte Carlo sampling techniques into iterative numerical linear algebra schemes. I will begin by describing how this approach can be applied to calculate ground-state electronic energies of small molecules. I will then describe the challenges encountered when applying this technique to calculate excited-state energies and describe our approach to addressing these challenges. Finally, I will introduce techniques inspired by existing quantum Monte Carlo methods that further reduce the cost of our approach and enable its application to larger molecular systems, such as a strongly correlated oxo-manganese(salen) transition metal complex.