If I've understood the hypotheses correctly, the covering relation gives a connected locally finite directed graph. The automorphism group is therefore a totally disconnected, locally compact group under the topology given by declaring the stabiliser of each point to be open; the point stabilisers are profinite. The automorphism group as a whole will be countable if and only if every point stabiliser is finite.
There's no reason for orbits of an automorphism to be finite: for instance, your poset could be $\mathbb{Z}$ and the automorphism could be $x \mapsto x+1$.