If you're an average 20 year old woman who doesn't get outside very much and has an iron deficiency, is taking 1000iU of Vitamin D safe?

Yes it is safe. But it may not be necessary, so do not waste your money unless you get checked first from a doctor who can check your Vitamin-D level in your blood which will likely be normal. But that is the best way to know if your body needs more of this vitamin or it has adequate amount for your health needs.