  In this paper, we prove the existence of $H^2$-regular coordinates on
Riemannian $3$-manifolds with boundary, assuming only $L^2$-bounds on the Ricci
curvature, $L^4$-bounds on the second fundamental form of the boundary, and a
positive lower bound on the volume radius. The proof follows by extending the
theory of Cheeger-Gromov convergence to include manifolds with boundary in the
above low regularity setting. The main tools are boundary harmonic coordinates
together with elliptic estimates and a geometric trace estimate, and a rigidity
argument using manifold doubling. Assuming higher regularity of the Ricci
curvature, we also prove corresponding higher regularity estimates for the
coordinates.
