In my town, and I suspect a lot of others, there are no regulations about restraining your own dog on your own land. Off your property, yes, there are leash laws. But in my town you have have your unleashed dog outdoors on your unfenced land all you want.
So fence, invisible fence, whatever. The dog owners are doing nothing wrong. And it certainly isn't rude to have your own dog outside on your own land.
Exactly who was this petition supposed to be turned into? The police? The town clerk? If there are no regulations about fences or tying up dogs, no one is going to accept the petition.
But I would warn the dog owners about the petition. As PPs have mentioned, this new family raising red flags for me. Invisible fences may keep dogs in their yards, but they don't stop humans from entering the yard and doing something to the dog.