This paper addresses the problem of parameter estimation in a quantized and online setting. A sensing unit collects random vector-valued samples from the environment. These samples are quantized and transmitted to a central processor which generates an online estimate of the unknown parameter. This paper provides a closed-form expression of the excess mean square error (MSE) caused by quantization in the high-rate regime i.e., when the number of quantization levels is supposed to be large. Next, we determine the quantizers which mitigate the excess MSE. The optimal quantization rule unfortunately depends on the unknown parameter. To circumvent this issue, we introduce a novel adaptive learning vector quantization scheme which allows to simultaneously estimate the parameter of interest and select an efficient quantizer.