Fast radio bursts (FRBs) are astrophysical transients of currently unknown origin, and so far several events have been detected at extragalactic distances. The dispersion measure (DM) of the radio signal is a probe of the integrated electron density along the line of sight and therefore allows to map the electron distribution within the large-scale structure. Since a fraction of electrons get expelled from galaxies by feedback, they are anticorrelated with halos at large scales and hence the angular DM correlations show a scale-dependent bias caused by primordial non-Gaussianity. Although the signal is weaker than in other probes like galaxy clustering, FRBs can potentially probe considerably larger volumes. We show that while studying the FRB clustering signal requires very large samples, correlations in the DM are cosmic-variance limited on large angular scales with only ∼103−4 events. A tomographic analysis of the angular DM correlation function can constrain the local primordial bispectrum shape parameter fNL to a precision down to fNL∼O(1) depending on assumptions about the FRB redshift distribution and the astrophysical feedback on large scales. This makes FRBs a competitive probe to constrain inflationary physics.