Note which colour button was clicked
I think in your case you can just do this:

document.body.style.background = colorChosen;


Avoiding all the ugly if else part.

A more robust solution will be using a lookup table to decouple the DOM ids from the proper colors string:

const colorFor = {
    "somethingRed": "red",
    "somethingBlue": "blue",
    "somethingBlack": "black"
};

...

document.body.style.background = colorFor[colorChosen];


With somethingBlaBla I mean a meaninful name that not contains the color.
The purpose is to contain the changes in case you want a different color in the future or if you want that more options have the same color.

Another improvement is to have a guard condition to avoid errors or weird situations:

var buttonClick = this.addEventListener ("click", function(e) {
    console.log(e.target.id + " button was clicked");
    var colorChosen = e.target.id;

    if (!colorFor[colorChosen])
        return false;

    document.body.style.background = colorFor[colorChosen];

});


Instead of just return from the function you can log an error or pup up a dialog with a warning to inform the user. 
Or you can reset to a default color the background.

In the same way you can use a style instead of setting the color on the DOM element.

This could be better in some situation, depends on your app code convention.