What WordPress handles for veterinary clinics
Online booking for consults and vaccinations. Booking plugins such as Amelia add services, times and booking forms; choosing a staff member is in its paid tiers.
After-hours emergency guidance. A tap-to-call phone link and a clear note about after-hours availability are ordinary page content.
New patient forms for pets. A form plugin can collect the basic details before a first appointment; keep sensitive information in a secure system chosen for it, not a website form or email. Owner contact details and pet records are personal information. Keep them in your booking or practice system rather than an email inbox, and ask only what the first visit needs.