This is a very classic sign of canine allergy, or atopy. Chewing at the feet usually means they are having a hypersensitivity reaction to something - can be anything from food to pollen to laundry detergent to insect bites and more.
Most of the time when the feet are that irritated a steroid is needed to alleviate the itch. You can try benadryl, 50 mg., three times daily for a few days and see if this helps relieve it at all.
Let me know if you have more questions,
Dr. Debbie
Veterinarian
Full time practicing companion and exotic animal veterinarian