Update: the dog was found! But here’s my story. I’ve been pet sitting for 2 years and was told these dogs were perfect with no issues. I locked the door the night before (I have proof) and woke up to one of the dogs missing. I searched the entire house and noticed the front door was open. I panicked, grabbed treats and her harness, and searched the neighborhood for 3 hours. There’s a camera pointing at the door, but nothing was caught. I informed the owners right away. I called all councils and vets, but the dog isn’t registered and had no collar. One owner threatened to sue me if the dog doesn’t come back, and I offered a full refund for both stays, but they declined. What can happen in this case? Can I get sued? Help 
Which door did you lock that you have proof of? Was it the front door? And what proof do you have?
It was the front door. There’s security camera footage of it being locked.
So if you KNOW there’s video of you locking the door, there must be footage of the door being opened. Maybe the cameras only pick up humans, not pets? Also, could one of the dogs open doors? I had a cat that could do this.
Document everything—your communications with the owners, your actions, and the lack of proof of the dog’s ownership. Consult a legal professional to understand your rights and prepare for any claims. Stay calm and keep searching. You did your best!