This just happened to me, actually. My cat was sick a few days ago and I brought her to the vet and she had a fever, probably from a vaccine but they did a blood test anyway. 2 days later, I was about to go do my walkings and my vet called saying that the blood tests were abnormal and she needed to come in right away. This is my cat I've had for over half of my life and I have her alone with nobody else to help, so I had to take her and stay with her.
I explained the situation to the client and dropped off her key, and Rover directly helped her find someone to cover for that day and the future. Rover also set my profile to away (I could change it back at any time) so even though my current bookings would stay active, I wouldn't appear in search results until I was ready to take new clients again.
The best thing to do is contact Rover and give them as much information as possible. If you don't want to talk about it to the client, they can do it for you, but really it's best if you do send them a message. I didn't really want to talk about it because I was upset but messaged her anyway in hopes that she would understand.