This would actually be great if I could rent a puppy for an hour for my kids or something. Maybe you should turn this into an AirBnB for pets. I'm sort of kidding, but sort of serious. I bet PETA would love you ;)
DAAS - Dogs As A Service (in the cloud of course). I'd be all over it - great chick magnets in the park :) I remember half-seriously looking for something like this a few years ago and found these guys: http://www.flexpetz.com
I think you can "rent" a puppy/dog for an hour at your nearest animal shelter. I think the people who work there would be more than happy if you walk a puppy/dog with your kids and give it some love for an hour.
Different animal shelters will vary on their policy, some being more open than others. For example a shelter where I volunteered put in a policy that volunteers had to commit to (something like) 15 hours/month. Needless to say volunteerism dropped significantly the next month...
Did you work at the luckiest shelter on the planet? The one I used to volunteer at [moved away] was usually clamoring for volunteers (at least in the summer).
But then you have to go to the animal shelter. What about a startup that brings the shelter puppy to you for a fee, which is then split with the shelter? Win Win Win?