Skeletal Maturity. A person stops growing when the growth plates in the bones close, which is medically known as "skeletal maturity." typically this is 16-18 for girls and 18-20 for boys. Bones in the foot generally mature about 2 years sooner than the long bones that affect height. Keep in mind that these are averages, so some people are on the very high or low ends of the spectrum.
Around 18. Most of the bones of the foot stop growing around the age of 18. By looking at an x-ray of the foot a doctor can tell if the bone has stopped growing. Some people may continue to grow a short time after 18 but that is the age for most people.