Oh @Grandma 
Is this sit through THS ?
I personally would not have taken the sit without that information. Was this a last minute sit ?
Has the homeowner given you the Welcome Guide ? The information should be in there .
Are you aware of the THS clause 5.3.5 that says the sitter has to pay for vet fees upfront ?
https://forum.trustedhousesitters.com/t/sitter-has-to-pay-veterinary-expenses-up-front
You do need to insist that the pet owner gives you the registered vets details and confirmation of how they will pay for any emergency treatment . If they refuse to do so , give them notice that you will be leaving in 24 hours to give them time to come back and look after their poorly pet or to make alternate arrangements for their care.
You should also advise member services of the situation so you can find a replacement sit.
Really not fair of an owner to leave you in this predicament of caring for a sick pet without disclosing this prior to your arrival
and without important information like the registered vet contact details.
For context, we have happily accepted and completed several sits with elderly pets; some that had medical conditions that were being managed with medication. In each case the owners were open and upfront with everything disclosed in the listing and their needs and care thoroughly discussed with us prior to confirming the sit . Including vet details and confirmation from the owner that they had an account set up so no fees were to be paid by us upfront.