How long can I wait with anemia to see if it goes away without seeing a doctor?

Hard to answer... You would need bloodwork to see how anemic you are. I would take iron sulfate with vitamin c atleast twice a day. Also, eat dark green, leafy vegetables. If you get dizzy easily or short of breast or have fatigue, you should see your doctor asap.
DO NOT WAIT. Anemia (low hemoglobin) is a sign of some other problem. Possible reasons for anemia range from the less severe and more easily addressed to life threatening diseases. You need to be seen so that the cause of your anemia can be determined and the appropriate care plan set up.
How do you feel? 2 questions determine this, how symptomatic are you and do you understand the cause of your anemia well enough to treat it. If low iron, you can try iron supplements 1st, with a good healthy diet. If from heavy cycles, are you treating that, like taking birth control pills and is it working? Unknown anemia is as hard to treat as asking a mechanic to fix your car with no information.