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?