Becoming a veterinary assistant doesn't require any specific education or degree program. This is an entry level position in veterinary medicine and most times vets will train new staff on the job. Be aware that a veterinary assistant position is different from that of a "veterinary technician" in many states. Veterinary assisting courses will not be sufficient to allow you to take the National Veterinary Technician Exam or be credentialed as a veterinary technician.
However, this lack of required training/education is due to the fact that there weren't veterinary assistant programs available in the past that were approved by an veterinary medical associations. There have been lots of online veterinary assistant programs for years who's curriculum and information hasn't been written, checked or approved by professionals in the field. Now there are at least a few veterinary assistant programs where the curriculum is written and approved by veterinarians and credentialed veterinary technicians. There are even a couple that count as college credits should you decide to go on with a degree in veterinary technology. If you wish to take an online certification program, choose one that is approved by a state or national veterinary medical association or through a college that also offers and AVMA approved veterinary technology program. These types of certification courses will be better recieved by potential employers.
But in general you can simply send your resume with all animal experience listed to local veterinary hospitals and try to get some interviews. Working as an assistant is a wonderful way to determine if this type of work is what you really want to do and if you should look into getting a degree in veterinary technology.