Fact is, women don't know and really don't care if you have approach anxiety or not. Women are too busy with men who are already approaching them then to be concerned with men who are just strangers and are not approaching them.
If you don't approach her, she will simply believe you don't like her and she will move on to make herself more approachable to men.
Most women's general rule towards men is, she will reciprocate and respond to men who talk to her, but she will rarely ever initiate a conversation with male strangers even if she is attracted to him. Further to this, women will rarely initiate anything from the initial date to activities during the date. So if you, as man do not initiate, she will simply assume, you are not interested. In the back of her mind, she may also believe this man is cowardly and weak and she isn't attracted to weak, cowardly men. So by not initiating, you may even cause her to be unattracted to you.
Once a woman is attracted to man, she will make herself available temporarily. Women will give man a small window of opportunity and he must pick up on that. If he does not within due time, the window shuts and never opens again.
As a man, it is your job, to recognize these opportunities and seize it before she moves on to a more proactive man.