12 weeks. At 12 weeks gestation, you can have an amniotomy done to take sloughed skin cells to test for genetic disorders. However, it is not a risk-free procedure, and you should discuss the risks with your obstetrician.
NIPP at 9 wk. There is an expensive but non-invasive form of prenatal testing called by the acronym NIPP(non invasive prenatal paternity) . It requires a sample of mothers blood and some from the designated dad. You can source this on line. The alternatives involving putting a needle up the birth canal or through the abdominal wall to get fetal tissue, or waiting until after the baby is born. .
Starting at 10 weeks. Fetal dna testing to screen for abnormalities can be done as early as 10-12 weeks by chorionic villus sampling(cvs). Cvs is usually done when: prenatal screening is abnormal; maternal age > 35, family history of genetic disease; prior chromosomal abnormalities with pregnancy loss. Or at 15-20 weeks an amniocentesis may be opted for, for the same indications. Discuss the risks with your obgyn.