I have an element like this.
click() is a function that's only defined on HTML elements. Fortunately it's a convenience function and we can implement it ourselves pretty easily i.e.
document.getElementById('box_w').dispatchEvent(new Event('click'));
That's all the click() function does underneath the hood anyway.
Here's an example:
document.getElementById('box_w').dispatchEvent(new Event('click'));
<g id="box_w" onclick="alert('hi')">
<rect fill="#FFFFFF" height="68.522" stroke="#FFFFFF" stroke-miterlimit="10" width="119.297" x="306.673" y="384.406"></rect>
</g>
Try to use onclick instead of click.
document.querySelector("g#box_w").onclick();
Remember that what we are doing here is calling the interface of the shape. So onclick needs to be defined in svg as it is in the question:
<g id="box_w" onclick="recordResponseKeyboard('W');"></g>