We demonstrate polarization-insensitive ultralow-power second-harmonic generation frequency-resolved optical gating (FROG) measurements with a fiber-pigtailed, aperiodically poled lithium niobate waveguide. By scrambling the polarization much faster than the measurement integration time, we eliminate the impairment that frequency-independent random polarization fluctuations induce in FROG measurements. As a result we are able to retrieve intensity and phase profiles of few hundred femtosecond optical pulses with 50 MHz repetition rates at 5.2 nW coupled average power without control of the input polarization.