This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian manifold $M$ of bounded geometry and positive injectivity radius, so that we have plenty of geodesically convex charts. Let $K\subset M$ be a regular closed subset (i.e. is equal to the closure of its interior) such that the boundary is rough -- say at best Lipschitz. For instance, take (the closure of) a corkscrew domain, an NTA domain, John domain etc. I'm even willing to just consider the interior corkscrew condition. Now I am interesting in applying results that would hold if $\partial K$ were contained entirely in a chart to 'larger' $K$. This means I need to cover $\partial K$ with geodesic balls, and in my setup I would like to take balls $B_i$ of half the radius (still covering) so that the closures of said smaller balls form a closed cover.
It is at this point my feeling for what might possibly go wrong fails me, and I have no idea if things can break at this point, due to bad behaviour of the intersection $\partial B_i \cap \partial K$, or what-have-you. As an example, this paper considers 'locally strongly Lipschitz domains', which have the pleasant property there exist transverse vector fields along the boundary, which I can see might ensure one could find 'nice' $B_i$ with 'nice' $\partial B_i \cap \partial K$. Ahlfors regularity seems to be prominent here.
To give a concrete idea of what I'm doing, I'm considering extension operators for functions on $K$ to functions on $M$, and hoping to work locally and then patch together by smooth partitions on unity. The trouble is, I don't know that, when I restrict to the local problem, I get something resembling a reasonable set under the chart map: it seems to me to depend on making sure I have charts whose boundaries intersect $\partial K$ in a nice way. Once I have that out of the way, I seem to be able to proceed without issue.
