We study theoretically the transfer of the light field orbital angular momentum (OAM) to propagating electrons upon photoemission from quantum well states. Irradiation with a Laguerre-Gaussian mode laser pulse elevates the quantum well state into a laser-dressed Volkov state that can be detected in an angular and energy-resolved manner while varying the characteristics of the driving fields. We derive the photoemission cross section for this process using the S-matrix theory and illustrate how the OAM is embodied in the photoelectron angular pattern with the aid of numerical calculations. The results point to a new type of time-resolved spectroscopy, in which the electronic orbital motion is addressed exclusively, with the potential for a new insight in spin-orbitally or orbitally coupled systems.