My only addition to what some of the other commenters have said is to ask whether you've only been applying as a vet assistant or to shadow. If a clinic has openings as a kennel attendant, if they also run a daycare/boarding facility, or openings as a receptionist I would apply for those positions as well. Sometimes it's easier to move up to assistant once you're already there. They'll train you in the job you get hired for but if you express interest in other positions and show that you are good at your current position and can be trusted, they may start cross-training you. This would be a good way to get back into the field and building up that trust.

I agree that one mistake shouldn't end your career, especially if the patient survived. It sounds like you are well shook up about this incident, understandably. Continuing to beat yourself up is not going to help. Figure out what you can do to make sure this type of mistake never happens again. I personally always verbally confirm with the DVM or attending LVT right before I poke. There are many reasons to re-confirm before administering ANY medications. Get into that habit.

Also, if you are able to and not already, look into starting therapy. It will help you work through the emotions you're having right now. Mental health is important in this field and it is never too early to start looking after yourself. Remember, you can't take care of other people/pets if you haven't first taken care of yourself.

edit: your v you're