Fast radio bursts (FRBs) are astrophysical transients of still debated origin. So far several hundred events have been detected, mostly at extragalactic distances, and this number is expected to grow significantly over the next years. The radio signals from the burst experience dispersion as they travel through the free electrons along the line-of-sight characterised by the dispersion measure (DM) of the radio pulse. In addition, each photon also experiences a gravitational Shapiro time delay while travelling through the potentials generated by the large-scale structure. If the weak equivalence principle (WEP) holds, the Shapiro delay is the same for photons of all frequencies. In case the WEP is broken, one would expect an additional dispersion to occur which could be either positive or negative for individual sources. Here, we suggest to use angular statistics of the DM fluctuations to put constraints on the WEP parametrized by the post-Newtonian parameter gamma. Previous studies suffer from the problem that the gravitational potential responsible for the delay diverges in a cosmological setting, which our approach avoids. We carry out a forecast for a population of FRBs observable within the next years and show that any significant detection of the DM angular power spectrum will place the tightest constraints on the WEP to date, Δγ < 10(-15).