When a motor does not run, we will need to find out which of the following components is/are not working: Laser controller, motor driver, motor, and wiring. One way is to change each component with spare parts. When we do not have spare parts, we can do a few tests.
Many machines has the same motor driver model for x and y, that makes the job easier. Check the rating of x and y motors and adjust the Amp setting on the drivers to avoid damage. Use low speed for testing because high speed can cause issue even when parameters are correct.
Let's say the x motor is having issue.
Swap the controller's x and y output. Now the y output controls the x motor driver. If left/right button can move the y motor, the controller signal and wiring between controller to the x motor driver should be good. If up/down button can move the x motor, the x motor, the x motor driver and wiring between the x driver and x motor are good.
There are only three major components along with the power supply to run a stepper motor. After some tests, we will find out the bad components eventually.