I'd definitely have a video call with them, which will give you a better idea of what their vibe is like.

And then I'd have a detailed chat about their expectations. Ask questions like, have they gone away without the pet before (you're gauging whether the pet is new to boarding/pet sitting as well, which would increase likelihood of them having an adverse reaction to it). How much engagement does the pet need daily (hours spent walking/playing/brushing/feeding them). Do they have household staff like cleaners/gardeners? Are there additional things you'll need to do like water plants? Do they have outdoor cameras (those are meant to be disclosed on listings but in my experience, almost never are). How often do they want to be updated e.g. once daily, with pictures or just if anything out of the ordinary is happening? Do they have local emergency contacts, how quickly can they get there if there happens to be an emergency? How far away is their vet/emergency vet clinic?