It seems to me that the answer is NO if $X$ is a DM stack.  If I'm not mistaken, it suffices to give a smooth finite type separated connected Deligne-Mumford stack over $\mathbb{C}$ which is simply connected (since such a thing has no non-trivial finite etale covers, let alone finite etale covers by a scheme).  But [this paper][1] of Behrend and Noohi shows that the weighted projective lines $\mathcal{P}(m, n)$ (constructed by taking the stack quotient of $\mathbb{A}^2\setminus\{0\}$ by the $\mathbb{G}_m$-action $\lambda\cdot(x,y):=(\lambda^m x, \lambda^n y)$) are simply connected.   The proof is easy; one just uses the long exact sequence for homotopy groups associated to the fibration $$\mathbb{G}_m\to \mathbb{A}^2\setminus\{0\}\to \mathcal{P}(m, n).$$

**Added later**: The answer seems to be no for algebraic spaces as well.  Example 5.7 [here][2] is simply connected if I'm not mistaken, and is not a scheme by Remark 3.4 in the same paper.


  [1]: http://arxiv.org/pdf/math/0504309.pdf
  [2]: http://projecteuclid.org/download/pdf_1/euclid.pja/1117805146