The poset consisting of a single countable antichain, with all elements incomparable, is [locally finite](http://en.wikipedia.org/wiki/Locally_finite_poset), but the automoprhism group consists of any permutation of the elements, which is an uncountable set.

Meanwhile, the automorphism group of any structure admits a natural topology, where the basic open sets are determined by a finite piece of the automorphism. That is, for any finite partial automorphism $p$ of the structure, one may consider the set of all automorphisms that extend $p$, and call this a basic open set. This topology is useful in diverse contexts, but perhaps you haven't really given us enough information about your context to determine if it might be useful for you.