How long will it take for my body to become anemic without iron?

It'd take longer than 3 months to become anemic. But you do need to buy a multivitamin with iron. Don't be lazy or cheap, this is your body you are talking about. And being anemic isn't fun. Being sick is awful. I am anemic and I feel like s h i t. So take care of yourself. Get some iron, and eat healthy and enough.