The gonality of a graph measures how difficult it is to move chips around the entirety of a graph according to certain chip-firing rules without introducing debt. In this paper we study the gonality of circulant graphs, a class of vertex-transitive graphs that can be specified by their number of vertices together with a list of cyclic adjacency relations satisfied by all vertices. We provide a universal upper bound on the gonality of all circulant graphs with a fixed adjacency list, which holds irrespective of the number of vertices. We use this upper bound together with computational methods to determine that the gonality of the \(4\)-regular Harary graph on \(n\) vertices is \(10\) for \(n\geq 16\). As a special case, this gives the gonality of sufficiently large antiprism graphs to be \(10\).