The availability and accuracy of Channel State Information (CSI) play a crucial role for coherent detection in almost every communication system. Particularly in the recently proposed cell-free massive MIMO system, in which a large number of distributed Access Points (APs) is connected to a Central processing Unit (CPU) for joint decoding, acquiring CSI at the CPU may improve performance through the use of detection algorithms such as minimum mean square error (MMSE) or zero forcing (ZF). There are also significant challenges, especially the increase in fronthaul load arising from the transfer of high precision CSI, with the resulting complexity and scalability issues. In this paper, we address these CSI acquisition problems by utilizing vector quantization with precision of only a few bits and we show that the accuracy of the channel estimate at the CPU can be increased by exploiting the spatial correlation subject to this limited fronthaul load. Further, we derive an estimator for the simple Quantize-and-Estimate (QE) strategy based on the Bussgang theorem and compare its performance to Estimate-and-Quantize (EQ) in terms of Mean Squared Error (MSE). Our simulation results indicate that the QE with few-bit vector quantization can outperform EQ and individual scalar quantization at moderate SNR for small numbers of bits per dimension.