How to build Non Linear Vector auto regression using Neural Nets
The type of artificial neural nets (ANN) you are looking for is ANN with cycles which is widely known as recurrent neural networks(RNN). As you have mentioned Vector Auto regression (VAR) I would recommend you to check out this paper which compares VAR to RNN on two different tasks pertaining to two different real world datasets.

One specific type of RNN is Long short term memory (LSTM). It is particularly good at learning over long sequences. Although RNN is also perfectly capable of that theoretically, but with little empirical success. So, in recent years LSTM has gained huge popularity and has been very successful over sequence learning tasks. I would recommend this book which provides a fair bit of understanding of ANN with cycles and the tasks they are being applied to. 

For implementation of RNN there are many APIs available. Keras provides a high level API for the same. 

For modelling multivariate time series with LSTM check out this tutorial. 

You can also have a look at this question that I answered on cross validated.