Generally, no. The idea that grain-free diets help with skin allergies is a common myth. When pets have food allergies, they are almost always allergic to a protein source (like chicken, beef, dairy, or lamb), not grains (like corn, wheat, or rice).
There's no scientific evidence to support grain-free diets as a treatment for most skin allergies. In fact, some grain-free diets have been linked to other health concerns in pets. If your vet suspects a food allergy, they will recommend a specific "hypoallergenic" or "novel protein" diet trial, which may or may not be grain-free, focusing instead on eliminating common allergens.