Make-on-demand combinatorial synthesis libraries (CSLs) like Enamine REAL have significantly enabled drug discovery efforts. However, their large size presents a challenge for virtual screening, where the goal is to identify the top compounds in a library according to a computational objective (e.g., optimizing docking score) subject to computational constraints under a limited computational budget. For current library sizes -- numbering in the tens of billions of compounds -- and scoring functions of interest, a routine virtual screening campaign may be limited to scoring fewer than 0.1% of the available compounds, leaving potentially many high scoring compounds undiscovered. Furthermore, as constraints (and sometimes objectives) change during the course of a virtual screening campaign, existing virtual screening algorithms typically offer little room for amortization. We propose the approximate-but-exhaustive search protocol for CSLs, or APEX. APEX utilizes a neural network surrogate that exploits the structure of CSLs in the prediction of objectives and constraints to make full enumeration on a consumer GPU possible in under a minute, allowing for exact retrieval of approximate top-k sets. To demonstrate APEX's capabilities, we develop a benchmark CSL comprised of more than 10 million compounds, all of which have been annotated with their docking scores on five medically relevant targets along with physicohemical properties measured with RDKit such that, for any objective and set of constraints, the ground truth top-k compounds can be identified and compared against the retrievals from any virtual screening algorithm. We show APEX's consistently strong performance both in retrieval accuracy and runtime compared to alternative methods.