Toklotursa,
The only shooting categories that require boots are the costume-based categories: Classic Cowboy and B-Western.
In any other category you can wear boots, shoes, moccasins, or even go barefoot. BUT, the operative rules are:
"Clothing must be typical of the late 19th century, a B-Western movie or a Western television series." (Shooters' Handbook, page 2) and
Outlawed Items: "All types of athletic shoes or combat boots, no matter the material from which they are constructed" and "Nylon, plastic, or Velcro accouterments." (Shooters' Handbook, page 4)
There is a difference of opinion on what footwear follows the rules and you will see the spirit of the rule stretched quite a bit (in my opinion). If you have a specific pair of boots/shoes you are considering wearing, you can post a picture and get some opinions.
AM
