What paperwork would I mail in with my taxes to show I had the foster children in my home for the entire year, when their birth parent already claimed them on taxes?

You will need a letter from the foster care agency showing that you were the foster parent and that the children were in your home the full year.  Your social worker should be able to provide this for you.  

There is also generally a case document assigning them to you which you may use as a supporting document.

Because your return has to be mailed and there is a dispute, it may take 4-6 months to process your refund.

First you need to establish you had the legal "custody" of the children.  The paper work from the foster agency.  You also can provide school and medical records to establish residency