I would ask the owner. It may be an awkward conversation, but I think if you phrase it as what would they like you to do if the dog gets sick/worse during your stay. Is there a cap on vet expenses, or an alternative contact you could reach while they're gone? If I'm going to be unreachable, I always leave a contact number for my parents or a close friend, so if any questions come up during the stay at least the sitter has someone to call for advice. I'm sure the owner has considered the possibility that the dog could pass away while in your care, so your question shouldn't be surprising.
I'm currently sitting a senior dog and as the owner dropped him off, he mentioned that should the dog get sick/start to decline while they are gone (2 weeks) to not pursue any life saving treatments and peacefully end his life. He also requested that I not call unless I really felt I needed to, as it would just upset his wife. My own dog is also a super senior, and I know I wouldn't want anyone spending thousands of dollars to keep him alive, so I do understand where the owner was coming from, but I personally would want to be notified and would tell my sitter that. Thankfully the dog is doing just fine 1 week into his visit! But should he start to decline, I plan to follow my gut and do what is best for the dog. He goes to the same vet my dog does, so I feel better knowing the staff there and that they won't pressure me or make me feel guilty for making a hard decision about an older dog. I will say, we didn't discuss what to do with the body if the dog passes away...I guess I hope the vet can hold it until the owner returns? I didn't even consider it until I read your question, but hopefully I never have to deal with it!