Here is what i found on another website
“Allergies is one of the most common causes of feet licking/chewing in dogs. Environmental and/or food allergies could be the underlying cause. I would speak with your veterinarian about screening for allergies and potentially starting a hypoallergenic diet for your dog if this is a consideration.
Pododermatitis, or foot infections, are also somewhat common in dogs. These infections are usually caused by a fungus (yeast) or bacteria in the environment. When it is humid out, these infections are more common, because they thrive in moist, humid environments. Dogs with hairy feet are predisposed to these infections if the hair is not kept short between their toes and pads. Your veterinarian will be able to tell you if this is the culprit causing the chewing.
Anxiety and boredom are the last two more common causes of dogs licking and chewing at their feet. Just like people, some dogs will chew on their nails when they are nervous or anxious. There are mild medications that can be used to decrease the amount of anxiety your dog experiences, and hopefully diminish the clinical signs. Talk to your veterinarian about this possibility as well. Nail biting is not normal and if it is anxiety-related, it would be good to slow the progression of the anxious behaviors while your dog is still young, because it can escalate as he/she gets older.
Boredom can also cause some nail biting and feet chewing behavior. These dogs are typically high energy dogs that need lots of exercise and stimulation. Kong toys and Nylabones are good for this type of dog, to keep them occupied when they are not interacting with other dogs or people. ”