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 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.

Surely there are simply connected algebraic spaces that are not schemes; but I don't know any examples (the standard Hironaka-type example of an algebraic space which is not a scheme has a finite etale cover by a scheme).

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 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 is simply connected if I'm not mistaken, and is not a scheme by Remark 3.4 in the same paper.

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 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.

Surely there are simply connected algebraic spaces that are not schemes; but I don't know any examples (the standard Hironaka-type example of an algebraic space which is not a scheme has a finite etale cover by a scheme).

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 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.

Surely there are simply connected algebraic spaces that are not schemes; but I don't know any examples (the standard Hironaka-type example of an algebraic space which is not a scheme has a finite etale cover by a scheme).