There are three things wrong with your code:
- It's getElementByIdnotgetElementByID(case-sensitive).
- You didn't convert the strings you get from the input to numbers. You can do this easily by adding a +in front of thedocument.getElementById('number1').value;
- You were doing comparisons against variables, not strings. Ex question == ADDinstead ofquestion == 'ADD'
See corrected jsFiddle example
document.getElementById('mainButton').onclick = function () {
    //Getting values of inputs and saving them to variables
    var numberOne = +document.getElementById('number1').value;
    var numberTwo = +document.getElementById('number2').value;
    //Setting values of the equations
    var addition = (numberOne + numberTwo);
    var subtraction = (numberOne - numberTwo);
    var multiplication = (numberOne * numberTwo);
    var division = (numberOne / numberTwo);
    //Prompting user for their desired equation
    var question = prompt("Do you wish to ADD, SUBTRACT, MULTIPLY, or DIVIDE?").toUpperCase();
    //If statement to show the proper equation based on the user's prior prompt
    if (question == 'ADD') {
        alert('I added the shit out of those numbers for you - turns out it is ' + addition);
    } else if (question == 'SUBTRACT') {
        alert('Did some of this, some of that, some minusing - your answer is ' + subtraction);
    } else if (question == 'MULTIPLY') {
        alert('Yeah, I multipled the numbers, big whoop, wanna fight abouddit? the answers over there --> ' + multiplication);
    } else if (question == 'DIVIDE') {
        alert('This ones my favorite, I love a good division - ' + division);
    };
};