This cookie recipe uses both milk and butter, so a lactose allergy would require substituting both of these ingredients.
Soy is a very common substitution for people with lactose allergies, but many soy products are produced in the same factories as milk, and may contain traces of “casein” which is a lactose protein. There are many other types of milk nowadays, below are the most common lactose-free milks:
The best way to substitute butter is to use margarine, which is made from vegetables rather than milk. There are many different types of margarine, so a small trial-and-error period may occur before you find the best margarine for the desired cookie texture.
Substitutions found from Kids With Food Allergies: A Division of the Asthma and Allergy Foundation of America
“Kids with Food Allergies.” Cooking and Baking Without Milk Ingredients. Kids With Food Allergies, n.d. Web. 25 June 2016. http://www.kidswithfoodallergies.org/page/milk-allergy-recipe-substitutions.aspx