There are very few mathematical results governing the interpolation of functions or their gradients on Delaunay meshes in more than two dimensions. Unfortunately, the standard techniques for proving optimal interpolation properties are often limited to triangular meshes. Furthermore, the results which do exist, are tailored towards interpolation with piecewise linear polynomials. In fact, we are unaware of any results which govern the high-order, piecewise polynomial interpolation of functions or their gradients on Delaunay meshes. In order to address this issue, we prove that quasi-optimal, high-order, piecewise polynomial gradient interpolation can be successfully achieved on protected Delaunay meshes. In addition, we generalize our analysis beyond gradient interpolation, and prove quasi-optimal interpolation properties for sufficiently-smooth vector fields. Throughout the paper, we use the words 'quasi-optimal', because the quality of interpolation depends (in part) on the minimum thickness of simplicies in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$. Furthermore, the current best mathematical estimates for minimum thickness have been obtained on such meshes. In this sense, the proposed interpolation is optimal, although, we acknowledge that future work may reveal an alternative Delaunay meshing strategy with better control over the minimum thickness. With this caveat in mind, we refer to our interpolation on protected Delaunay meshes as quasi-optimal.