This paper studies a novel way to estimate the position of an object in an indoor environment, using the Channel State Information (CSI) that a Visible Light Communication (VLC) system collects to maintain the link-level connectivity. First, supervised learning is applied to characterize, the effect that an object in variable but known positions has on the received optical wireless signal. Second, the trained classifier is used to estimate the new unknown positions that the object may take, making use of the instantaneous CSI that is used to equalize the data-carrying signal samples in reception. The practical validation of the proposed positioning approach was done with the aid of a software-defined VLC link based on OFDM, in which a copy of the intensity modulated signal coming from a Phosphor-converted LED is captured by Photodetectors (PDs) in different room locations. Then, the CSI of the VLC receiver is used to train a Random Forest classifier, which will predict the position of the object during the assessment phase. The performance evaluation of our experimental setting shows that the proposed VLC-based positioning approach can reach a few centimeter accuracy, provided that a proper training is executed, without the necessity of deploying a large number of PDs in the room, or adding a VLC receiver on the object to be tracked.