Blood work during first trimester to find out illness on fetus?

Andrea • Got my first BFP at 14dpo (vvvfl 🔝) on cycle #2 for baby #1. Baby is due on October 6th 2017! ❤
What's the name of that test some women do to try and find out any DNA abnormality or possible illness the baby may have/develop? Also, how soon or late can you do it? I will be 10 weeks and 2 days and would like to get it done but I don't have an OBGYN so I'm trying to find out here and go try to get the test done. 
PS. I'll be having a home birth with a midwife and doula. They don't prescribe this kind of test.