While Vernon Center, with its population density of 89, does not have any pet-friendly hotels, you can find plenty just outside in the cities of Vernon, Verona, and New Hartford. For your peace of mind, there is a vet clinic just a mile down the road in Vernon.
Right behind the local post office -- hidden from all but the local's eyes -- is Vernon Park, a little hole-in-the-ground with big wonders that await you and Fido. Here, you can unwind and rejuvenate from your daily stresses in the natural beauty of the upstate mountains. Right beyond the playground equipment lies miles of majestic trails along a flowing river that leads to multiple scenic shallow waterfalls just waiting for Fido to jump and play in. Pack a light lunch and some water bottles and a bowl, and stop at one of the benches along the trail to enjoy a picnic.
While you are in Rome, check out the Erie Canalway Trail. This trail has the official designation of being a National Recreation Trail and is a 36-mile-long route that extends from Dewitt (near Syracuse) to Rome. Along this trail, you'll find opportunities for you and Fido to picnic, canoe, hike, and play in the canal. Dogs are required to be on a leash to be able to walk around the canal. Parking starts at the old Erie Canal Village, and from there, you can access the beginning of the trail and take as long (or as short) of a walk as you like while you and Fido enjoy the natural wooded terrain and wildlife of upstate New York!