Abstract: The extension of Standard Model made by inclusion of additional U(1) gauge L-mu – L-tau symmetry can explain the difference between the measured and the predicted value of the muon magnetic moment and solve the tension in B meson decays. This model predicts the existence of a new, light Z' vector boson, predominantly coupled to second and third generation leptons, whose interaction with electrons is due to a loop mechanism involving muons and taus. In this work, we present a rigorous evaluation of the upper limits in the Z' parameter space, obtained from the analysis of the data collected by the NA64-e experiment at CERN SPS, that performed a search for light dark matter with 2.84 x 10(11) electrons impinging with 100 GeV on an active thick target. The resulting limits touch the muon g – 2 preferred band for values of the Z' mass of order of 1 MeV, while the sensitivity projections for the future high-statistics NA64-e runs demonstrate the power of the electrons/positron beam approach in this theoretical scenario.