I've seen this film 3 times now, and I think it's fairly certain that there is nothing within its text that substantiates a confirmed answer to this.
The first time I saw Calvary, I assumed it was Inspector Stanton: and furthermore, this would be foreshadowing his reveal as Father Lavelle's eventual killer. I largely based this assumption on their conversation regarding the Webley:
Stanton: If you were to tell me you needed the gun for, say, you're dog not doing so well. You're not sure he's going to last and you wouldn't want him to suffer...
However, once you've watch the film again (with the perspective of knowing who the culprit is), you begin to interpret the behavior of the players differently.
Stanton is also (with the exception of the bar-keep Brendan Lynch [who does have a motive to distribute vengeance against Lavelle by this point]) the only resident not to attend the church fire. He also demonstrates disdain for whomever the culprit of the fire was, calling them out for their lack of originality:
Stanton: Sure, any Eejit can start a fire.
This would, to me, indicate that he considers the burning of a church to be an insubstantial measure of retribution.
Stanton is also a practicing homosexual, although unlike Leo he never openly declares this to be the result of abuse at the hands of a priest as a child, but the concept is discussed in his presence.
Whilst it would appear outwardly that he has no motive to kill Father Lavelle's dog, the point of the film is that no-one has a real reason to hurt Lavelle, other than his manifest association with the institution that has seen to have caused more harm than good.
It is for this reason that, anyone could be the true culprit, as the entire town has seemingly turned against this good man because of the failures of the wider Church. There are so many elements of the community that have completely 'lost faith', I see the dogs murder as a deliberate comment on the inevitability of Lavelle's downfall: that there was at least 2 people capable of performing such extreme acts in the sake of retribution.
For me, Calvary seemed like a macabre accompaniment to Father Ted: both of which are effective at representing the current perception of Catholicism in Ireland, albeit very differently.
Niamh Connolly: I hope this island isn't some hideaway for paedophile priests.
Ted: well Niamh, we're not all like that, say there is 200 million priests in the world and 5 per cent of them are paedophiles, thats
still ...only 10 million .
Sins of the father...