I've figured out what was troubling me.
Input.GetAxisRaw("Vertical")
and Input.GetAxisRaw("Horizontal")
were fine the whole time. Somehow, there was some code in the script that in the Editor worked just fine, but when running as standalone gave me a calculation equal to 0 (it depends on the Delta Time, so I assume that the Delta Time when running compiled is much smaller than when running in the Unity Editor) and thus the character didn't move at all.