Extracting embeddings of Categorical Variables
As far as I understand, usually embeddings are initialized with random values. If there are pretrained embeddings, they can be loaded, but of course there are no such things for categorical variables.

So if you simply create an embedding of a categorical feature, you'll get a vector representation, but it will have random variables. So at first you'll need to train the neural net. After it is trained, you could take the output of the embedding layer and use it in regression model.

But are you sure that you need the embedding? You could do mean encoding or frequency encoding.