  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 gets 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 $\sim 10^{3-4}$
events. A tomographic analysis of the angular DM correlation function can
constrain the local primordial bispectrum shape parameter $f_\mathrm{NL}$ to a
precision down to ${f_\mathrm{NL}}\sim \mathcal{O}(1)$ 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.
