Neural networks work on any data with consistent dimenionality.
If your images all have the same size, than the HOG features will also have consistent dimensionality over all images.
Indeed it is quite standard to train NNs on complexer features like HOG or SIFT instead of on the raw pixel data.
As a deeper point, HOG can be computed using NNs. In fact, the HOG computation can be seen as a specific instance of convolutional neural networks, albeit with some obscure connectivity restrictions and obscure kernel functions (like atan, and normalization functions).
This is a purely theoretical point, however - I would not recommend implementing HOG using NNs!