I do not believe that there is any way, based on these facts, that Pease in a Pod can possibly have grounds to allege discrimination. There is nothing in these facts to suggest discrimination and there is nothing to suggest that the owner or anyone else connected to the restaurant is part of a group of people protected against discrimination.
Discrimination occurs when a person is treated differently because of their membership in a given group. There are laws that ban discrimination against people on the basis of their race, their age, their disability status, and many more attributes. The question as stated gives us no reason to believe that the state is refusing to give her a liquor license because she is a woman or because she is a member of any other group.
Theoretically, the owner might be able to argue that her restaurant is being discriminated against because it is a vegetarian restaurant. However, there is no evidence to show that other restaurants that are similar to hers, but not vegetarian, have been granted liquor licenses. This would make it hard to prove discrimination.
Discrimination has to be based on the group membership of the victim. It would have to be about some attribute of the owner or because of the vegetarian nature of the restaurant. You cannot successfully claim discrimination simply because you think that a law has been misinterpreted in your case. Without some evidence that she and her restaurant are being treated differently than similar establishments, there is no reason to think that she has a credible claim of discrimination.