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](https://ell.stackexchange.com/a/57392/6700) and [here](https://ell.stackexchange.com/a/59247/6700).

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By the way, a well-known precedent for "take a walk" occurs in [this song](https://www.google.com/search?client=safari&rls=en&q=%22take+a+walk+on+the+wild+side%22+song&ie=UTF-8&oe=UTF-8&gws_rd=ssl).