Most vet clinics advertise in state veterinary publications. For example if your state veterinary medical association publishes a magazine then they will often have classified ads listed. Other sites to look at would be those that cater specifically to the veterinary field---veterinary magazines, veterinary websites, etc. There is one called "where techs connect" that lists jobs, however I don't know how active that site is lately.
Veterinary clinics MAY list jobs in your local newspaper, but that's not as common as it used to be. They may also list on the online job hunting websites but you may have to pay for a membership or register to see job listings with those.