How do I use a string method inside a string prototype extension method

In javascript this is function scoped, so creating a new function creates a new this.

Your forEach call has a callback, which is a function, and within that function this is no longer the string, but most likely the window

The solution is to simply store a reference to whatever this is in the outer function

String.prototype.includesAny = function(search) {
    var found = false,
        input = this;
    search.forEach(function(str) {
        if (input.toLowerCase().includes(str.toLowerCase())) {
            found = true;
        }
    });
    return found;
};

Array.forEach also has an optional thisArg that can be used

String.prototype.includesAny = function(search) {
    var found = false;

    search.forEach(function(str) {
        if (this.toLowerCase().includes(str.toLowerCase())) {
            found = true;
        }
    }, this);
    return found;
};

Or even better, use Array.some

String.prototype.includesAny = function(search) {
    return search.some(function(str) {
        return this.toLowerCase().includes(str.toLowerCase());
    }, this);
};

As a sidenote, extending native prototypes is generally a bad idea.

You're probably correct about this not being a string because you are in the foreach. So store an instance of this in a variable before entering the foreach and use that instead.

var stringToCheck = this;
search.forEach(function...

You need to save your this in other variable to use it in other functions scope.