This behavior is not normal, since dogs are not herbivorous. Many times they do it due to a digestive system discomfort/disorder (colitis, gastritis, flatulence, digestive indiscretion, etc.). It could also be due to parasites, nutritional deficiencies, among others. A veterinarian can help you find the reason.
And the answer is no, you should not allow your dog to eat grass. Grass has sharp edges that can damage the delicate structures of the walls of the digestive tract. In the simplest case, grass can be contaminated with fertilizers, insecticides, parasites, etc.