If your question is "if I have a Deligne-Mumford stack where all points have trivial automorphism group, is it isomorphic to any coarse moduli space for that stack?" then the answer is yes. As Kevin says, a D-M stack with no automorphisms is just an algebraic space, and the map to a coarse moduli space, by definition, is universal amongst maps to algebraic spaces.
I'll note, if this seems too easy, this is because assuming that a particular moduli problem has a solution given by a D-M stack is asking a lot. It's not necessarily an easy thing to prove.