Yes, anxiety can cause numbness. When you are experiencing a panic attack, you can get numbness and tingling of the arms and legs, which are also called paresthesias. The mechanism for getting numbness when you have anxiety can occur in two ways.
In the first way, you can get numbness in your body when you have stress and anxiety, and this in turn activates the fight or flight response. In the fight or flight response, there is a shunting of blood from the skin to the skeletal muscle, to prepare the body for fighting or fleeing. The shunting of blood from the skin can then be felt as numbness.
In the second way, you can get numbness in your arms and legs if you hyperventilate. When you have anxiety, you might breathe faster, and the faster you breathe, the more shallow your breaths. So as you breathe faster, you are not oxygenating your body, as your breaths are now shallow. At the same time, you are blowing off alot of carbon dioxide from your body with each rapid breath, which further worsens the hyperventilation. So with hyperventilation that you may get with anxiety, you can feel numbness and tingling of your extremities as you are not getting enough oxygen to your body.