I'm unaware of any cordless tool that will drive those staple nails; they're a hammer-in type of fixing. We can order a wire stapler, but the staples are very fine and unsuitable for fencing. Also, these staplers are typically air driven, so you'd need a compressor close by.
Remember to have your posts on the other side of the fence to your dogs. You should only need to place a staple at the top and bottom and a couple in the middle. Are you planning on running a guide wire and tension strainers, as these will increase the rigidity of your fence and minimise the number of staples you'll need?