The website should have a bio, maybe some suggested care routines (like preferred deworming protocols), and also information about the best way to contact the vet. If there’s a regular schedule, like they are in a certain area on certain days of the week, I’d include that (though you have to keep it up to date). Emergency coverage and options may also be appropriate.
For social media, I think it’s great to humanize your people. Showing pictures of interesting cases or cute animals or something of community interest can be nice. Always be careful with client confidentiality and also… in a world where vets are overworked and underpaid, there’s a line to walk to not make things either too gritty or too cheerful.
Social media is a good way to remind people to schedule teeth and vaccinations, again a nice picture of animals receiving those services may do well to get people thinking about them. Ask the vet what it is they wish more clients knew/did- maybe their practice area more people need to know about nutrition, or teeth, or safety, whatever. You can use it to remind people of practices that are helpful in extreme weather, or what information you want them to gather before they call.
Social media can do two things for you: remind and communicate with existing clients, and also flesh out the people for potential clients choosing between veterinary practices. Finally, if you want to raise their profile outside the local area for some reason, this is also a way.
My vet was asked to x-ray a beached whale carcass which made excellent social media, and also sometimes shares photos of healthy baby animals they’re seeing, always a winner.