In his notes: http://math.uga.edu/~pete/243integrals1.pdf (Wayback Machine, Pete Clark outlines an axiomatic approach to the Riemann Integral. He doesn't use the language of sheafs, but it seems implicit in his definition before Theorem 1. He goes on to show that the fundamental theorem of calculus follows, and then that such an integral exists.
I'm a little shaky on sheaf-theoretic language (I haven't been taught this material formally), so I just want to run the following reformulation past any experts out there to make sure its correct.
Given a sheaf $F$ on $\mathbb{R}$ containing the sheaf of bounded real-valued continuous functions on X, can the Riemannian integral be defined axiomatically as:
$I: F \rightarrow \mathbb{R}$ such that
i. $f,g \in F$ and $f\le g \implies If\le Ig$
ii. $f,g \in F$ and $supp(f) \cap supp(g)$ is a point or empty, then $I(f+g)=If+Ig$
iii. $I(C)=c(b-a)$ where $C$ is the constant function on the interval $(a,b)$ with value $c$.
This then begs the question as to whether the Lesbegue integral can be also defined in a similar fashion, say by replacing $\mathbb{R}$ by some measurable space, the sheaf of continuous real-valued functions by the sheaf of simple functions, and iii. by the usual Lesbegue integral of an indicator function where now $I$ is positive homogeneous.