Your height of 5 feet (60 inches) and age of 13 is completely normal for both boys and girls. While diet can affect your health, final adult height is primarily determined by genetic factors (how tall are your parents and other family members?).
A well-planed vegetarian diet can provide you with all the nutrients your body needs. This is essential as you enter adolescence because your body undergoes one of the most significant growth spurts since infancy. If you are committed to being a vegetarian, you may want to meet with a dietitian who could review your diet and make sure you are getting the nutrients you need. The dietitian will respect your choice not to eat meat and will advise you which foods are essential to a healthy vegetarian diet. You should also let your pediatrician know about your lifestyle choice. He/she can also offer helpful suggestions that will help you be successful.