  We prove $\varepsilon$-regularity theorems for varifolds with capillary
boundary condition in a Riemannian manifold. These varifolds were first
introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first
variation control for all such varifolds (and free-boundary varifolds
generally) satisfying a sharp density bound and prove that if a capillary
varifold has bounded mean curvature and is close to a capillary half-plane with
angle not equal to $\tfrac{\pi}{2}$, then it coincides with a $C^{1,\alpha}$
properly embedded hypersurface. We apply our theorem to deduce regularity at a
generic point along the boundary in the region where the density is strictly
less than $1$.
