Can I as a regular citizen, no. If law enforcement has enough evidence they can acquire a subpoena and force you to take one. Also some states can request one for child custody cases. I had a friend who ex denied the baby was his so the state told him he would have to pay or prove he wasn't the father with a DNA test. (DNA proved he was the father)
You cannot force someone to get DNA testing. He or she must be willing to have the test done. This is because there are some samples that are needed from that particular person and can only be gotten if he or she is willing.
In our state you can by going to the D.H.S. offices or C.P.S..Also by calling the circuit clerk at court house and asking for the # for the child support enrorcement offices. They will issue a court order for D?a test as the 1st step to collecting child support. Here in Arkansas my neighbor just got done doing this w/her kids dad. They were never married and he was and is living in California. It took almost a year because of all the state lines the order had to cross.