Relativistic jets are believed to be born magnetically dominated.
Very and ultra-high energy cosmic rays can be efficiently accelerated by magnetic reconnection in these sources. We here demonstrate this directly, with no extrapolations to large scales, by means of three-dimensional relativistic magnetohydrodynamical (3D-RMHD) simulations of a Poyinting flux dominated jet. We inject thousands of low-energy protons in the region of the relativistic jet that corresponds to the transition from magnetically to kinetically dominated, where its magnetization parameter is $\sigma \sim 1$. In this region, there is efficient fast magnetic reconnection which is naturally driven by kink instabilities (KI) in the initial helical magnetic fields of the jet. We find that the particles are accelerated by Fermi process in the reconnection regions (and by drift in the final stages) up to
energies $E \sim 10^{18}$ eV for background magnetic fields $B \sim 0.1$ G, and $E \sim 10^{20}$ eV for $B \sim 10$ G. We have also derived directly from the simulations the acceleration rate due to magnetic reconnection which has a weak dependence on the particles energy
$ r_{acc} \propto E^{-0.1}$, characteristic of exponential growth. The energy spectrum of the accelerated particles develops a power-law tail with spectral index $p \sim -1.2$. This hardness of the spectrum must decrease when particle losses and feedback into the background plasma are included. Our results can explain observed flux variability in the emission of blazars at the very high energy band as well as the associated neutrino emission. Successful applications of our results to the blazars MRK 421 and TXS 0506+056 are also discussed.