There's nothing wrong with the patterns of "take a run", "make a run", and "make a walk", but each specific phrase may or may not have precedents that describe going to the store. Specifically:
He made a run to the store. [This is the normal way to say it!]
He took a run to the store. [This sounds strange. Usually "take a run" means to run for exercise, not to perform a chore.]
He made a walk to the store. [This sounds very strange. There is no precedent for "make a walk" to mean walking. So, when you combine "make" and "walk", the listener is forced to think that "walk" means something like a concrete sidewalk, which you could make. If you say "make a walk to the store", your meaning is clear, but it sounds foreign.]
He took a walk to the store. [This sounds strange in the same way that "take a run to the store" sounds strange. Normally, if you "take a walk", you are walking for exercise or you are quitting something abruptly. You might "take a walk around the block", but there's little precedent for "take a walk to the store".]
There is no grammatical rule for any of this. You just have to learn from experience how new phrases echo familiar phrases. Some more information about how phrases play a role in English comparable to individual words is here and here.
By the way, a well-known precedent for "take a walk" occurs in this song.
