To make a run usually means to travel to a nearby location, buy something or perform an errand, then typically return." It implies it's done rather quickly - typically for the purpose of getting one or a few items that are needed right away (it's a run so it's supposed to be quick).
E.g. you might make a run to pick up some soda for a party starting in an hour (and might even call that a soda run), but you wouldn't make a run to do your monthly shopping with your mother-in-law. That's more of a trip - though "make a trip" can be used to describe the same things as "make a run."
Anyway, unless you want to come off as being sarcastic, this doesn't work with walk. Make a walk is not a well-known expression - a walk is not something that's made when it is performed.
Take a walk does mean to go walking leisurely for a time, and take a walk to X does mean to walk to X, often implying that its for leisure. When you make a run, you are NOT travelling for leisure (your purpose is to get something and come back), you are travelling for a well-defined purpose, so take a run does not work to express the same thing.
Now, take and make are "low-level" English words that can be used to express many things, albeit without precision. Make can often mean something similar to perform, to bring about, or to manifest and take can often mean something similar to consume.
So there may be situations where a speaker might be compelled to say take a run - if the "run" is something that can be "consumed". E.g. - let's say the speaker is at an event where there are limited opportunities to run - "taking a run" would be appropriate to say there - "Hey, you can go take your run now" or "Hurry up and take your run before you lose your spot."
And there is also situations where the act of walking could be performing - it would not be out of place for a fashion model's boss or manager to say "It's time to make your walk now" to tell them to stroll down the catwalk.
But make a run and take a walk are well-known phrases and you can't switch run and walk between them without sounding awkward to a native speaker.