Fast Radio Bursts (FRBs) are extra-galactic radio transients which exhibit a distance-dependent dispersion of their signal, and thus can be used as cosmological probes. In this article we, for the first time, apply a model-independent approach to measure reionization from synthetic FRB data assuming these signals are detected beyond redshift 5. This method allows us to constrain the full shape of the reionization history as well as the CMB optical depth $\tau$ while avoiding the problems of commonly used model-based techniques. 100 localized FRBs, originating from redshifts 5-15, could constrain (at 68% confidence level) the CMB optical depth to within 11%, and the midpoint of reionization to 4%, surpassing current state-of-the-art CMB bounds and quasar limits. Owing to the higher numbers of expected FRBs at lower redshifts, the $\tau$ constraints are asymmetric (+14%, -7%) providing a much stronger lower limit. Finally, we show that the independent constraints on reionization from FRBs will improve limits on other cosmological parameters such as the amplitude of the power spectrum of primordial fluctuations.