If the verb for “photograph” is take, I presume that the verb for a “photographic copy” should also be take.
The word photocopy is often abbreviated to copy. I have noticed the verb make is used for copy.
Considering the above, which is used more: to make a copy or take a copy?
When you photograph something, you take a photo.
When you copy something, you make a copy.
When you copy something photographically and use the portmanteau word photocopy to describe the process (and its result), you are using the noun-as-adjective photo to describe the noun copy.
Because copy is the chief element in this combination, it requires the same verb to describe the copying process as you would use for copy alone, namely make: so you make a photocopy.