We report on a search for elementary particles with charges much smaller than the electron charge using a data sample of proton-proton collisions provided by the CERN Large Hadron Collider in 2018, corresponding to an integrated luminosity of 37.5 fb$^{-1}$ at a center-of-mass energy of 13 TeV. A prototype scintillator-based detector is deployed to conduct the first search at a hadron collider sensitive to particles with charges ${leq}0.1e$. The existence of new particles with masses between 20 and 4700 MeV is excluded at 95% confidence level for charges between $0.006e$ and $0.3e$, depending on their mass. New sensitivity is achieved for masses larger than $700$ MeV.