Welcome to Rover! Congratulations on booking your first stay. It sounds like a long one, too!
If it were me, I would send them a regular update then transition into mentioning that it looks like their pup has a skin problem on their leg. I would say "skin problem" because I'm not a vet, and you can cause problems making assumptions, even when it's super obvious what the problem is. I'd include it in a regular, upbeat update to show them that I'm not freaking out about it, just informing them and seeking direction. Then you can insert your thoughts and give them some options for what you can do for their pup. Then WAIT to see what they say. I agree with Walt - never, ever give a dog medicine or put medicine on them without written permission from the owner beforehand.
Something like this is what I would go with:
"Hey Sara - I hope you and the family are having a wonderful trip. Rex is doing great today, as usual. We went for a hike down to the pond and got back right before it started raining, so we've just been chilling on the couch watching TV together this afternoon. It looks like Rex is having a skin issue on his back, left leg. I'm not a vet, but it looks like the hot spots my dog gets from time to time. I have hot spot medication if you want me to try that, or I can take him in to see the vet, or I'm happy to do whatever you suggest. Just let me know. He scratches it a little bit, but otherwise he's his usual, goofy self and having a great day."
I hope that's helpful. Keep up the good work!