I made a quick demo of something that you hopefully should be able to base your code on, using the Pointer Lock API.
I forked this pointer-lock-demo repo and modified it to add a random movement element.
Here is the link to my GitHub page: https://aristocrates.github.io/pointer-lock-demo
And here is the link to my repo: https://github.com/aristocrates/pointer-lock-demo
The javascript code of importance is contained in app.js, in the canvasLoop(e) method.
The only thing I changed from the original demo was after the lines
x += movementX * 2;
y += movementY * 2;
I added two lines to represent random movement:
x += Math.floor(Math.random()*3 - 1);
y += Math.floor(Math.random()*3 - 1);
There are still plenty of things you could improve, but hopefully this can help you get started.