This letter proposes a data-driven input reconstruction method from outputs (IRO) based on the Willems' Fundamental Lemma. Given only output measurements, the unknown inputsestimated recursively by the IRO asymptotically converge to the true input without knowing the initial conditions. A recursive IRO and a moving-horizon IRO are developed based respectively on Lyapunov conditions and Luenberger-observer-type feedback, and their asymptotic convergence properties are studied. An experimental study is presented demonstrating the efficacy of the moving-horizon IRO for estimating the occupancy of a building on the EPFL campus via measured carbon dioxide levels.