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The poset consisting of a single countable antichain, with all elements incomparable, is locally finite, but the automoprhism group consists of any permutation of the elements, which is an uncountable set.

But you also said "connected". If by this you mean that the poset is linearly ordered, which is also commonly described as the connectedness axiom, then the poset must be either a finite linear order, the integers, the positive integers or the negative integers, and each of these posets has a trivial automorphism group.

If instead you mean that the graph relation underlying the poset is a connected graph, then consider an infinite antichain with a single top point above them all. This is still locally finite, since all intervals have at most two elements, and it is connected as a graph, but the automorphism group is again any permutation of the elements of the antichain, which is uncountably many automorphisms.

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.

The poset consisting of a single countable antichain, with all elements incomparable, is locally finite, but the automoprhism group consists of any permutation of the elements, which is an uncountable set.

But you also said "connected". If by this you mean that the poset is linearly ordered, which is also commonly described as the connectedness axiom, then the poset must be either a finite linear order, the integers, the positive integers or the negative integers, and each of these posets has a trivial automorphism group.

If you mean that the graph relation underlying the poset is a connected graph, then consider an infinite antichain with a single top point above them all. This is still locally finite, since all intervals have at most two elements, and it is connected as a graph, but the automorphism group is again any permutation of the elements of the antichain, which is uncountably many automorphisms.

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.

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The poset consisting of a single countable antichain, with all elements incomparable, is locally finite, 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.