I think this is true.

Take a smooth atlas $U\to X$, and notice that sections of $\mathcal{O}_X$ (which you can see as morphisms of quasi-coherent sheaves $\mathcal{O}_X \to \mathcal{O}_X$) correspond to sections of $\mathcal{O}_U$, such that the two restrictions to $U_1:=U\times_X U$ by means of the two projections to $U$ coincide.

This is also true of morphisms out of $X$: a morphism $X\to T$ where $T$ is a scheme corresponds to a morphism $U\to T$ such that the two compositions $U_1\to U\to T$ coincide (basically because $Hom(-,Spec R)$ is a sheaf, as you said).

This reduces the question to the fact that morphisms $U\to Spec R$ such that the two compositions $U_1\to Spec R$ coincide correspond to morphisms $R\to \mathcal{O}_U(U)$ such that the two compositions $R\to \mathcal{O}_{U_1}(U_1)$ coincide.

categoryof morphisms $X \to \mathbb{A^1}$. – Martin Brandenburg Oct 29 '11 at 22:53