Why is it "take a leak?"

It is because one takes an action, leak or piss denote actions, and the noun forms of these actions ended up using take instead of other available verbs.

To "leak," meaning to "make water" or piss, was first a verb. Shakespeare, Henry IV part 1:

Why, they will allow us ne'er a jordan, and then we leak in your chimney; and your chamber-lie breeds fleas like a loach.

And piss, as a verb, goes back to Middle English. Chaucer, the Parson's Tale:

An hound, whan he comth by the roser or by othere beautees, thogh he may nat pisse, yet wole he heue vp his leg and make a contenaunce to pisse.

By the 20th century, both words could also serve as nouns, denoting the action of leaking or pissing, according to the Oxford English Dictionary. At the same time, they acquired a periphrastic use in the phrase to take a leak:

(Tropic of Cancer, 1934) I stood there taking a leak.

(Heartless, 1934) There were puddles of sludge from the mud of the road, the waste water of the saloon, and any number of passing drunkards who thought to stop and take a piss on their way through.

Why take? Basically, take in this usage emphasizes the following noun. When verbs do this, they are called delexical or empty because the verb is less important than the following noun. To take a piss => to piss. To take a leak => to leak. It's not that anything is being literally taken, as with other meanings of to take, but rather that the verb introduces an action. Here is how the OED explains take as a delexical verb that emphasizes carrying out the following action:

81.a. To make, do, perform (an act, action, movement, etc.); to carry out. Often take forms with the object a phrase which is a periphrastic equivalent of the cognate verb: e.g. to take a leap is equivalent to to leap, to take a look to to look, to take one's departure to to depart, etc.

"have, v." has virtually the same entry (22). It's hard to answer why take was the verb and not, say, have. Without direct evidence (which would be very hard to come by for common periphrastic forms), I can only guess that it's an accident of use.

Disclaimer: I am far from a scholar in the English language.

I had always understood usage of the word "take" in these situations to simply be a shortened version "take time out/off for", as in:

"Take time off for a vacation", "Take time out for a nap", etcetera...

Another way to think of this would be the fact that what is being inferred as taken would be "time from your day/week/life"...