Have you looked at the  deformation theorem for  rectifiable currents? This essentially states that any integral current  $S$  can be approximated by a polyhedral current situated not very far from  $S$. Your smooth chain defines an integral current.    A good place to look  for more details is Frank Morgan's **Geometric Measure theory. A Beginner's Guide**,  Section 5.1. I believe that the strategy used in the proof of the deformation theorem could be useful for your problem too, or at least the weaker version suggested by Matthias Kreck.

More precisely, the  deformation theorem indicates that your chain $c$ can be approximated  (in various   norms on the space of currents) by a nice polyhedral chain $c'$, whose support   can be chosen in an arbitrarily small neighborhood of the support of $c$.       In particular $c'$ is homologous to $c$, if $c$ is closed.