Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is a (1+epsilon) scaled copy of P that contains S?

I do not see how scaling could give such a property. Consider an annulus in the plane (remove a part of it to make it a Jordan domain and also make it smooth if you want). Scaling any polygon inside the annulus by $1 + \epsilon$ makes also the "missing ball" inside the annulus larger. 

