A vector sensor, a type of sensor array with six collocated antennas to measure all electromagnetic field components of incident waves, has been shown to be advantageous in estimating the angle of arrival and polarization of the incident sources. While angle estimation with machine learning for linear arrays has been well studied, there has not been a similar solution for the vector sensor. In this paper, we propose neural networks to determine the number of the sources and estimate the angle of arrival of each source, based on the covariance matrix extracted from received data. Also, we provide a solution for matching output angles to corresponding sources and examine the error distributions with this method. The results show that neural networks can achieve reasonably accurate estimation with up to 5 sources, especially if the field-of-view is limited.