The child is ten years old. I have never seen the child and I wasn't in the state when the child was born

You request one though the court, the court will set up for approved labs for you to go to. You go to testing facility in your state, they go to testing facility in their state, all the tests are sent to the lab and a test is done.

Who is requesting the test?

In order to be recognized a Court-accredited lab MUST do the test (as FrChuck said).

You go to a lab in your State; the child is tested in a lab in his/her State.

Whether you were in the City when the child was born is immaterial. It's where you were when he was conceived that matters.