In this paper, we present a calibration algorithm for acoustic vector sensors arranged in a uniform linear array configuration. To do so, we do not use a calibrator source, instead we leverage the Toeplitz blocks present in the data covariance matrix. We develop linear estimators for estimating sensor gains and phases. Further, we discuss the differences of the presented blind calibration approach for acoustic vector sensor arrays in comparison with the approach for acoustic pressure sensor arrays. In order to validate the proposed blind calibration algorithm, simulation results for direction-of-arrival (DOA) estimation with an uncalibrated and calibrated uniform linear array based on minimum variance distortion less response and multiple signal classification algorithms are presented. The calibration performance is analyzed using the Cramér-Rao lower bound of the DOA estimates.