Electric field in a capacitively coupled, non-self-sustained RF discharge nitrogen plasma with external ionization generated by high-voltage ns pulses has been measured by ps Electric Field Induced Second Harmonic (EFISH) generation. The measurements are made both in the bulk of the plasma and in the sheaths, using the absolute calibration by the Laplacian field between two plane electrodes. The results are compared with the kinetic modeling calculations. The RF electric field in the sheaths is significantly higher compared to that in the plasma, due to the displacement of the electrons by the drift oscillations and the resultant plasma self-shielding. However, the kinetic modeling predictions indicate that the electron impact ionization in the sheaths is largely ineffective, due to the low electron density. The reduction of the electric field in the plasma by the self-shielding in the sheaths is moderate, such that the energy is coupled to the plasma by the below-breakdown RF field. The peak RF field in the plasma is in the range of 15-25 Td, indicating the efficient vibrational excitation of N2 by electron impact. The modeling predictions suggest that the targeted vibrational excitation of molecular species in a non-self-sustained RF discharge with external ionization is scalable to high pressures, electron densities, and discharge powers. The present approach can be extended to the vibrational excitation of other molecular species where the vibrational relaxation is relatively slow, such as CO, CO2, and H2.