I assume that by Sobolev topology you mean the topology induced by the Sobolev norm. Since all normed spaces are metric spaces the affirmative answer to your question follows from the fact that all metric spaces are paracompact. See e.g.
*A new proof that metric spaces are paracompact* by Mary Ellen Rudin ([pdf][1]).

*edit*

I missed that you are asking for connections modulo gauge equivalence. In that case I can refer you to Theorem 2 of [arXiv:1012.3180][2] where it is proved* that the moduli space of all connections is locally Hausdorff Hilbert manifold and that the space of all connections forms a $\mathrm{Gau}$-principal bundle over it. 

.* In the setting of Lie algebroids and their connections which subsume many classical cases.


  [1]: http://www.ams.org/journals/proc/1969-020-02/S0002-9939-1969-0236876-3/S0002-9939-1969-0236876-3.pdf
  [2]: https://arxiv.org/abs/1012.3180