In the weapon, change the value for the field Weapon - Allowed Movement to Moving.
If you want the weapon to be able to fire in any direction without the unit turning, change the field Stats - Arc to 360, this allows for the unit to acquire targets in any direction. Additionally, you will need to make the weapon have a turret with a Yaw Arc value of 360, allowing it to target enemies in any direction.
Also, the field in the turret called Idle dictates how it works. If it is set to Reset If Moving, you will get an effect similar to the Phoenix/Siege Tank (the difference is that Pheonixes can shoot while moving, Tanks can't).