If $n$ is unconstrained, as it seems to be in the question, then the polygon can be taken to be an $\epsilon$-accurate approximation to the optimal figure, and we can ask directly what that figure (or figures) look like.
If you don't assume the polygon is convex, the answer is trivial, in that you can get as close as desired to a collection of $k$ disjoint circles connected by very thin tubes. In this version of the problem the only constraint is that $A > k \pi R^2$ from which the maximum $k$ is easily determined.
If convexity is required, finding the tightest configuration of circles -- the one whose convex hull is a minimum-area figure containing $k$ or more discs of given size -- is a hard nonlinear optimization problem which is a variant of the classical problem of circle packing (finding the smallest circle enclosing $k$ unit discs). Recent experiments indicate that the optima, even for large $k$, do not have a connectivity pattern approximating the optimal lattice packing.
If the polygon is convex and you are satisfied with an asymptotically optimal solution, start from a hexagonal packing of circles of radius $R$ in the whole plane, and draw a circle of area $A$ that encloses as many of these as possible, then take the convex hull of the circles inside, then approximate the convex hull closely enough by a polygon.
(added: for asymmetry in high density finite packings up to n=348, see http://arxiv.org/abs/1002.0604 and a long series of theory papers by the same authors. Best known packings of small numbers of disks in circles, hexagons, squares, and other shapes are displayed at: http://www2.stetson.edu/~efriedma/packing.html .)