I don't have a separate contract exactly. I have a pre-stay survey that I make all my clients fill out before I book them or before I start a stay. In that survey, they have to agree to certain statements like "I understand that it is my responsibility to provide everything my pet(s) needs, and I will pay for any food that the sitter is required to purchase if food runs out. I will be informed in advance and receipts will be provided." Personally, I wouldn't require an owner to agree that they would pay for any damages to my home simply because they have no control over whether their dog damages my home. During the years I did dog boarding (before we moved to a new rental), I always felt it was my responsibility to make sure a dog doesn't damage my home. Plus, I think I would lose potential clients if I forced them to sign that contract, given the fact that I would never sign that kind of contract with a sitter.
To prevent damages, I just made sure I had a crates (or the owners provided one) that the pups could stay in while I wasn't home. Someone was home 24/7 when we were doing dog boarding. I had WAY more problems with adult dogs than puppies, but still no damages to speak of.
If you really want a contract, which might make sense if you're super packed with clients and you're not worried about losing some, you could probably find doggie daycare contract templates or liability waivers online, then introduce it to a couple clients at a time and see how they respond and if they book/rebook you. You could create a survey through a site like SurveyMonkey (the free one I use) with some statements they have to agree to in order to book the stay, and mix in some questions about their dog so it doesn't sound quite as formal. I set up different "collectors" for the same survey so each client is sent a unique link to the survey so I know that person agreed to the statements in the survey. I also ask for their contact info, their vet, emergency contact in case I can't reach them, and lots of dog questions.
I hope this was helpful. Good luck!